document resume se 016 300 secondary schools curriculum ... · behavioral objectives for geometry,...

DOCUMENT RESUME

ED 077 740 SE 016 300

AUTHOR Rogers, Arnold R., Ed.; And Others. TITLE Secondary Schools Curriculum Guide, Mathematics,

Grades 10-12, Levels 87-112.INSTITUTION Cranston School Dept., R.I.SPONS AGENCY Office of Education (DREW), Washington, D.C. Projects

to Advance Creativity in Education.PUB DATE 72NOTE 145p.; Draft Copy

EDRS PRICE MF-$0.65 HC-$6.58DESCRIPTORS *Behavioral Objectives; Calculu, Computers;

*Curriculum; Curriculum Guides; !ometry; MathematicsEducation; Number Concepts; Objectives; ProbabilityTheory; *Secondary School Mathematics

IDENTIFIERS ESEA Title III; Objectives Bank

ABSTRACTBehavioral objectives for geometry, algebra, computer

mathematics, trigonometry, analytic geometry, calculus, andprobability are specified for grades 10 through 12. Generalobjectives are stated for major areas under each topic and arefollowed by a list of specific objectives for that area. This workwas prepared under an ESEA Title III contract. (DT)

FILMED FROM BEST AVAILABLE COPY

Secondary Schools

U S DEPARTMENT OF HE AL TNEDUCATiON & WELF ARENAT.ONAL INSTITUTE OF

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CURRICULUMGUIDE

Cranston School DepartmentCranston, Rhode Island

1972

MATHEMATICSGrades 10-12 Levels 87-112

Secondary School

CURRICULUM GUIDE

DRAFT COPY

Prepared By

a curriculum writing team

of secondary teachers

Project PACESETTER

Title III, E. S. E. A., 1965

Cranston School Department

845 Park Avenue

Cranston, R.I. 02910

1972

ACKNOWLEDGEMENTS

Dr. Joseph J. PicanoSuperintendent of Schools

Mr. Robert S. Fresher Mr. Josenh A. murravAssistant Superintendent Assistant Sunerintendent

Dr. Guy N. DiBiasio Mr. Carlo A. sambaDirector of Curriculum Director of (rant Programs

Mr. Robert A. BerlamDirector, Project Pacesetter

Mr. Arnold R. RogersCoordinator, Pacesetter

Dr. John TibbettCurriculum Consultant

The final revision of the auides was completed during theProject Pacesetter Curriculum workshop, June 1972 bv:

Arnold R. Rogers, Chief EditorJoseph R. Rouleau-Production Snecialist

Eileen Sibielski-Conv Snecialist

Darnell McCauley Thomas RasgalloEmily Nickerson William Reilly

Robert Winsor

Typing by: Debra Santurri and Deborah Olivo-CHSW students

The basic material was contributed by:

C. Abosamra, S. Aiello, J. Alfano, S. Alfano, A. Ball, R.Bernier, A. Blais, E. Blamires, W. Campbell, S. Co.inors, R.Coogan, L. Corbin, J. D'Ambra, M. DeBiasio, A. DeLuca, S.Dilorio, T. Donovan, E. Fink, R. Forrest, J. Fricot, A.Gambardella, E. Geigen, D. Grossi, P. Gustafson, J. Herne,T. Lucas, A. Mangili, D. McCauley, W. Meciunas, M. Murnhv,H. Nicholas, E. Nickerson, M. Pitocchi, T. Rasnallo, A.Rogers, E. Sibielski, C. Snacagna, C. Spirito, P. St. Jean,E. Sullivan, R. Winsor.

PREFACE

The following levels consist of instructional objectives andactivities for each course of study within every curriculum area.These materials were produced by a staff of teachers working oncurriculum teams for Project PACESETTER. They are, therefore,the product of the experience of the professionals who will putthem to use.

This curriculum guide provides each teacher with curriculummaterials organized into behavioral objectives with a scope andsequence. The guide is intended to encourage feedback so that afully classroom tested curriculum will eventually result from theparticipation and suggestions of all teachers in the secondaryschools of Cranston.

OBJECTIVES IN TERMS OF LEARNING VARIABLES

Bloom and his collegues devised a taxonomy of educationalobjectives designed to classify the behavior of students in threedomains as a result of having participated in a series of in-structional experiences. The three domains are the cognitive(intellectual), the affective (emotional), and the psychomotor(physical). Within each of these domains there is a hierarchywhich denotes increasing complexity of learning which is shownbelow.

Cognitive

knowledgecomprehensionapplicationanalysissynthesisevaluStion

Affective Psychomotor

receivingrespondingvaluingorganizationcharacterization

frequencyenergyduration

The objectives which appear in these Curriculum guides havebeen stated in behavioral or performance terms. In addition to the

.general technique of the behavioral statement, the authors werecareful to differentiate the levels at which given behaviorscould be expected of the student. Thus, in the cognitive domain,a student's performance in the display of knowledge of a conceptis less complex than the student's performance when he appliesthe concept in a given situation. Similarly, in the affectivedomain, a response to a situation is not as complex as the displayof a value toward a given situation.

In initial classroom trials of this curriculum teachers willevaluate the appropriateness of the objectives and make recommenda-tions for revising, deleting, or adding to the objectives or ac-tivities.

LEVELS, OBJECTIVES, AND ACTIVITIES

The curriculum guides providet; here are organized into be-havioral objectives which generally include two major components.The first is the objective statement which specifies the behav-ioral variable--the intended behavior of the students as a resulto: having participated in a set of instructional experiences,the content or topic and the evaluative criterion which is some-times implicit in the behavioral objective. Curriculum writershave made every effort to classify the intended behaviors inkeeping with the work of Bloom and others. The objectives,then, are stated in terms of specific behaviors which rangefrom the simple, such as memorizing or translating, to the mostcomplex, such as synthesizing or evaluating. The second majorcomponent is comprised of activities which outline what thestudent should do to attain the objective. These activitiesare suggested and should be added to, deleted, or modified bythe teacher according to the needs and characteristics of in-dividual students and the teacher's own experience and knowledge.

It is important to note here that the objectives serve thepurpose of helping each teacher select appropriate learning ex-periences, communicate to others what is expected, and provideboth student and teacher with a standard for evaluating progress.Objectives should not be seen as limiting teacher innovation orwhat the student is expected to know.

Each of the curriculum areas is divided into major topicsor "Levels." Each level begins with a level objective which isfollowed by numbered objectives subordinate to it. Suggestedactivities follow each of these speclic objectives and arenumbered consecutively throughout the level.

EVALUATIVE CRITERIA

Many of the evaluative statements included in the behavioralobjectives are teacher orient d; final decisions on evaluationhave traditionally been the prerogative of the teacher. As wemove toward continuous progress and, eventually, individualizedinstruction, it is hoped that the evaluation component increas-ingly becomes the shared responsibility of both teacher andstudent.

TABLE OF CONTENTS

MATHEMATICS

Grader 10-12

Title Level*Suggested

Grade

Geometry, Block I M 87 10-11

Geometry, Block II M 88 10-11

Geometry, Block III M 89 10-11

Geometry, Block IV M 90 10-11

Algebra II, Block I M 91 10-11

Algebra II, Block IJ M. 92 10-11

Algebra II, Block III M 93 10-11

Algebra II, Block IV M 94 10-11

Advanced Geometry M 95 10-12

General Mathematics M 96 10-12

Modified Algebra,Part I M 97 10-12

Modified Algebra,Part II M 98 10-12

Computer Mathematics M 99 10-12

The Slide Rule M 100 10-12

Introduction to BasicCollege Mathematics M 101 12

Review Mathematics,Algebra M 102 12

Review Mathematics,Geometry M 103 12

Title

Advanced Algebra and

Level*Suggested

Grade

Trigonometry I M 104 12

Advanced Algebra andTrigonometry II M 105 12

Advanced Algebra andTrigonometry III M 106 12

Advanced Algebra andTrigonometry IV M 107 12

Advanced Mathematics,Trigonometry M. 108 12

Advanced Mathematics,Analytic Geometry M 109 12

Advanced Mathematics,Calculus M 110 12

Advanced Mathematics,Probability- M 111 12

Calculus andAnalytic Geometry M 112 12

* pages are numbered within levels only'

M 87

LEVEL OBJECTIVE

D. 1

THE STUDENT WILL DEMONSTRATE AN INCREASE IN HIS ABILITY TOAPPLY THE GEOMETRIC CONCEPTS OF SETS, INDUCTION, DEDUCTION,AND ANGLE RELATIONSHIPS BY COMPLETING THE FOLLOWING LEVELTO THE SATISFACTION OF THE TEACHER.

I. Sets and Set Relations

Objective #1: The student will increase in comprehen-sion of the basic elements of sets andset relations by completing the follow-ing suggested activities to the satis-faction of the teacher.

Activities:

The student will be able to:

1. Identify a well-defined set by the roster anddescriptive methods.

2. Describe sets as being finite or infinite.

3. Distinguish between the empty set and the setcontaining the element O.

4. Define subset and to differentiate and illustrateproper subsets, improper subsets, overlapping setsand disjoint sets.

5. Find the union, intersection, and complement ofspecified sets.

6. Use Venn diagrams to illustrate the operationsamong related sets.

7. Identify the set of real numbers and its subsets.

8. Explain the concept of one-to-one correspondencebetween the real numbers and points on a line.

9. Define absolute value and show its application re-garding distance between points. . ..

10. Explain the symbols used in comparing real numbers.

M 87 o. 2

11. Recognize and interchange various forms of rationalnumbers.

II. Induction and Deduction

Objective #2: The student will increase his ability tothink critically by applying principlesof induction and deduction in the follow-ing suggested activities to the satisfac-tion of the teacher.

Activities:


12. Explain the meaning, importance, and limitationsof inductive thinking.

13. Apply_ induction to a problem situatior in orderto reach a conclusion.

14. Recognize intuition as a tool in gaining insightin problem situations.

15. Explain the meaning, importance, and limitatio.nsdeductive thinking.

16. Apply deduction to a problem situation in orderto reach a conclusion.

17. Recognize deduction as a system used in reachingconclusions.

18. Constrast among intuition, inductive and deductivethinking.

III. Nature of a Deductive System

Objective #3: Tho student will apply the basic con-cepts of geometry to formulating defin-itions, postulates, and theorems by com-pleting the following suggested activ-ities to the satisfaction of the teacher.

Activities:


M 87 P. 3

19. Explain the statement that point, line, and planeare being described and not defined and they con-stitute the three basic undefined terms of a Eu-cledian geometric system.

2u. Explain the statement that precise definitionswould involve the introduction of less familiarterms which would in turn have to be defined.

21. Recognize a good definition.

22. Interpret postulates and theorems as they apply toa deductive system.

?3. Demonstrate a knowledge of the basic terms of be-ginning logic by indicating correct conclu ::ions,converses, contrapositives, inverses, etc.

24. Apply postulates and theorems concerning points,lines, and planes.

25. Apply the ideas which relate to the correspondenceof the set of real numbers and the set of pointson a line.

26. Apply the Betweenness of Points definition.

27. Recognize that a segment has exactly one mid-point.

IV. Angle Relationships

Objective #4: The student will apply definitions, postu-lates, and theorems to proofs involvingangles and angle relationships by complet-ing the following suggested activities tothe satisfaction of the teacher.

Activities:.


28. Apply the olgle measurement postualte and the pro-tractor postulate.

29. Explain that the protractor postulate does for an-gles and angle measures what the ruler postulatedoes for segments and lengths.

M 87 p. 4

30. Apply the angle addition theorem which will be usedextensively in proofs.

31. Apply theorems related to angles and angle relation-ships.

32. Apply those theorems and definitions related toperpendicular lines and right angles.

33. Apply those theorems and definitions related tocomplementary, supplementary, and vertical angles.

34. Write a complete demonstration of a two-columnproof.

M 88 p. 1

LEVEL OBJECTIVE

THE STUDENT WILL APPLY THE DEFINITIONS AND THEOREMS OF PAR-ALLEL LINES, PARALLEL PLANES, POLYGONS, AND CONGRUENCE TODEVELOPING TECHNIQUES OF PROOF BY COMPLETING THE FOLLOWINGLEVEL TO THE SATISFACTION OF THE TEACHER.

I. Parallel Lines and Planes

Objective #1: The will apply definitions andthe 3 ,ovolved in working with par-allel lines and planes and to developtechniques for proving examples involv-ing them by completing the followingsuggested activities to the satisfactiolof the teacher.

Activities:


1. Apply definitions as reasons for steps in proofsand as generators of algebraic equations in orderto solve number problems.

2. Recognize and identify by name all angle pairsformed by two lines intersected by a transversal.

3. Give measures of all angles formed when two paral-lel lines are cut by a transversal providing oneof those angles is known.

4. Prove whether or not two lines are parallel whengiven the measures of various angles formed by twolines cut by a transversal.

5. Recognize when auxiliary lines are necessary tocomplete a proof and justify their use.

6. Realize that not all statements can be proven di-rectly.

7. Recognize when a statement cannot he proven direct-ly and prove the statement by means of an indirectproof.

M 88 p. 2

II. Parallel Lines Applied to Polygons

Objective #2: The student will apply the concepts ofparallel lines and planes to the studyof polygons by completing the followingsuggested activities to the satisfactionof the teacher.

Activities:


8. Distinguish between polygons and other types ofgeometric figures as well as between various typesof polygons.

9. Recognize and identify various types of trianglesand quadrilaterals.

10. Apply parallel lines to proving various propertiesof triangles and quadrilaterals.

11. Apply properties of triangles and quadrilateralsto proofs.

12. Apply special properties of triangles and quadri-laterals to the solution of number problems.

13. Identify an exterior angle of a triangle and under-stand its relationship to the remote interior an-gles of the triangle.

14. Explain the properties of a polygon dealing withthe sum of its interior and exterior angles.

15. Find the measure of an interior or exterior angleof a regular polygon.

III. Congruence

Objective #3: The student will apply the various casesof congruence to proofs involving congru-ent triangles by completing the followingsuggested activities to the satisfactionof the teacher.

M 88 P. 3

Activities:


16. Recognize and name correspondences between twogeometric figures.

17. Interpret the concept of congruence.

18. Apply all ways of proving triangles congruent:SAS, ASA, AAS, SSS and the additional methods forright triangles: HA, HL, LL, LA. (SSA and AAAare never cases of congruence)

19. Define congruence of triangles as an equivalencerelation.

20. Recognize congruent triangles when they are over-lapping and prove them congruent.

21. Apply congruent triangles to prove that segmentsare congruent and angles are congruent.

22. Apply congruent triangles and corresponding partsto demonstrate special properties of triangles.

23. Apply congruent triangles and corresponding partsto demonstrate properties of special triangles.

M 91 p. 1

LEVEL OBJECTIVE

THE STUDENT WILL INCREASE IN COMPREHENSION OF THE STRUCTUREOF MATHEMATICAL SYSTEMS PARTICULARLY THE REAL NUMBER SYSTEM,THE ELEMENTS OF LOGIC, AND THE COMPUTATION WITH ALGEBRAICEXPRESSIONS BY COMPLETING THE FOLLOWING LEVEL TO THE SATIS-FACTION OF THE TEACHER.

I. Real Number System

Objective #1: The student will increase in comprehen-sion of the real number system by dis-tinguishing among the various categoriesof the real number system by completingthe following suggested activities tothe satisfaction of the teacher.

Activities:


1. Distinguish between types of numbers.

2. Interpret properties exhibited by particular setsof numbers.

3. Perform operations using particular sets of numbers.

4. Describe the structure of the Real Number System.

5. Distinguish between periodic and non-periodic deci-mals.

6. Express periodic decimals as the quotient of twointegers.

II. Logic

Objective #2: The student will apply the basic elementsof mathematical logic to given statementsand proofs by completing the followingsuggested activities to the satisfactionof the teacher.

Activities:


M 91 p. 2

7. Apply inductive reasoning.

8. Translate notation used in mathematical reasoning.

9. Apply deductive reasoning.

10. Distinguish between induction and deduction.

11. Form the negation of a statement.

12.. Form basic composite statements (conjunction, dis-junction, conditional, and biconditional).

13. Assign and organize in table form the truth valuesof composite statements.

14. Form negations of composite statements.

15. Form the variants of a -onditional.

16. Explain the nature of a tautology.

17. Interpret the nature of a deductive system.

18. Apply a deductive system to proofs.

III. Mathematical Structure

Objective #3: The student will increase in comprehen-sion of the structure of mathematicalsystems by completing the following sug-gested activities to the satisfaction ofthe teacher.

Activities:


19. Identify unary and binary operations.

20. Describe the concept of a group.

21. Identify properties exhibited by a group.

22. Describe the concept of a commutative (Abelian)group.

M 91 p. 3

23. Identify properties exhibited by an Abelian group.

24. Describe the concept of a field.

25. Identify the properties exhibited by a field.

26. Recognize the set of real numbers as a field andspecifically, as a complete ordered field.

IV Algebraic Expressions

Objective #4: The student will apply the principles ofalgebraic expressions to computation with-in the following suggested activities tothe satisfaction of the teacher.

Activities:


27. Evaluate algebraic expressions.

28. Simplify expressions using grouping symbols.

29. State the order of algebraic operations.

30. Perform the operations of addition and subtractionusing algebraic expressions.

31. Perform the operation of multiplication using thefollowing algebraic expressions:

a. monomial X monomial,

b. monomial X binomial,

c. binomial X binomial,

d. monomial X multinomial, and

e. multinomial X multinomial.

32. Perform the operation of division using the follow-ing algebraic expressions:

a. multinomial monomial,

b. Long Division, and

M 91 p. 4

33. Factor algebraic expressions such as the following:

a. common monomial factoring,

b. common binomial factoring,

c. difference between two perfect squares,

d. perfect trinomial square,

e. general trinomial, and

f. an+ bn.

34. Perform the expansion of binomials.

35. Perform the square of a trinomial.

36. Simplify algebraic fractions.

37. Perform the addition and subtraction of algebraicfractions.

38. Perform the multiplication of algebraic fractions.

39. Perform the division of algebraic fractions.

40. Simplify complex fractions.

41. Perform operations involving positive integralexponents.

42. Perform operations involving negative integralexponents.

43. Perform operations involving fractional exponents.

44. Translate fractional expenents to radicals.

45. Express a radical in simplest form.

46. Perform operations with radicals.

47. Simplify expressions with fractional exponents.

M 92 p. 1

LEVEL OBJECTIVE

THE STUDENT WILL APPLY THE PROPERTIES OF RELATIONS AND FUNC-TIONS AND METHODS OF SOLVING EQUATIONS TO A VARIETY OF PROB-LEM SOLVING SITUATIONS BY COMPLETING THE FOLLOWING LEVEL TOTHE SATISFACTION OF THE TEACHER.

I. Solutions of Open Sentences in One Variable

Objective #1: The student will increase in comprehen-sion of problem solving involving opensentences in one variable by completingthe following suggested activities tothe satisfaction of the teacher.

Activities:

The student will able to:

1. Solve a first-degree equation in one variable.

2. Distinguish between a conditional equation and anidentity equation.

3. Apply the operation laws for inequalities to thesolution of first-degree inequalities in on vari-able.

4. Graph the solutions of open sentences on a numberline.

5. Solve composite open sentences and graph their sol-utions.

6. Solve open sentences involving absolute value.

7. Solve quadratic equations by factoring.

8. Solve quadratic equations by completing the square.

9. Solve the general quadratic equation ax2 + bx + c0 by completing the square.

10. Solve quadratic equations by using the quadraticFormula.

M 92 p. 2

11. Interpret the discriminant to determine the natureof the roots of a quadratic equation.

12. Write a quadratic equation from its given roots bythe method of the sum and product of the roots.

13. Solve quadratic inequalities.

14. Solve irrational equations and identify extraneousroots.

15. Solve rational ewticns and identify extraneousroots.

16. Apply the techniques of solving linear ant quadrat-ic equations to problem solving.

17. Analyze and organize= given information in a verbalproblem in order to form an appropriate open sen-tence.

18. Solve different types of verbal problems such as:age, coin, consecutive integer, digit, geometry,mixture, motion, and work.

II. Relations and Functions

Objective #2: The student will increase in comprehen-sion of the concepts of functions andrelations by completing the followingsuggested activities to the satisfactionof the teacher.

Activities:


19. Define a relation.

20. Graph a relation on a Cartesian coordinate system.

21. Define a function.

22. Differentiate between a relation and a function.

23. Graph a function on a Cartesian coordinate system.

M 92 p. 3

24. Form the inverse of relations and functions andcompare their graphs.

25. Use functional notation.

26. Form and graph the sum, product, difference, quo-tient, and composition of functions.

III. Linear Functions

Objective #3: The student will apply the concepts oflinear functions to equations and theirgraphs by completing the following sug-gested activities to the satisfaction ofthe teacher.

Activities:


27. Recognize the equation of a linear function.

28. Determine whether a line is vertical, horizontal,or oblique.

29. Define the slope of a line.

30. Write the equation of a straight line in:

a. point-slope form,

b. :lope-intercept form, and

c. standard form.

31. Recognize that the slopes of parallel lines areequal.

32. State the relationship between the slopes of perpen-dicular lines.

33. Apply the distance formula.

34. Apply the midpoint formula.

M 93 p. 1

LEVEL OBJECTIVE

THE STUDENT WILL APPLY GRAPHING AND OTHER EQUATION SOLVINGTECHNIQUES INVOLVING CONIC SECTIONS, MORE THAN ONE VARIABLE,EXPONENTS, AND LOGARITHMS BY COMPLETING THE FOLLOWING LEVELTO THE SATISFACTION OF THE TEACHER.

I. Conic Sections

Objective #1: The student will apply the properties ofconic sections to graphing these sectionsand solving equations by completing thefollowing suggested activities to thesatisfaction of the teacher.

Activities:


1. Understand the formation of the four sections of aright circular conical surface.

2. Identify the equations of a circle, parabola, el-lipse, and hyperbola.

3. Graph the four conic sections at the origin.

4. Translate the four conic sections to the point(h, k).

5. Solve and graph linear-quadratic systems of equa-tions.

6. Solve and graph quadratic-quadratic systems of equa-tions.

7. Solve and graph systems of quadratic inequalitiesin two variables.

8. Apply quadratic systcms to the solution of verbalproblems.

II. Solutions of Equations in more than one Variable

Objective #2: The student will apply graphing and theproperties of equations to the solutionof various systems of equations by com-pleting the following suggested activ-ities to the satisfaction of the teacher.

M 93 p. 2

'Activities:

The ?tudent will be able to:

9. Classify and identify systems of two simultaneousequations in two variables.

10. Find solution sets of systems of simultaneous line-ar equations in two variables graphically.

11. Find solution sets of systems of simultaneous line-ar equations In two variables algebraically:

a. by the multiplication-addition method, and

b. by the subtraction method.

12. Find solution sets of the general system of twoequations in two variables.

13. Solve algebraically systems of three equations inthree variables.

14. Solve algebraically and graphically systems of onefirst-degree equation and one second-degree equa-tion.

15. Solve algebraically and graphically systems of twosecond-degree equations.

16. Solve algebraically and graphically systems of in-equalities.

III. Exponential and Logarithmic Functions

Objective #3: The student all apply the exponential andlogarithmic functions to the solution ofarithmetic and word problems by completingthe following suggested activities to thesatisfaction of the teacher.

Activities:


17. Distinguish between algebraic and transcendentalnumbers.

M 93 p. 3

18. Recognize and graph an exponential function.

19. Define a logarithm.

20. Identify and graph a logarithmic function.

21. Recognize exponential and logarithmic functions asinverses.

22. Interchange exponential and logarithmic notation.

23. Apply exponential laws to properties of logarithms.

24. Apply the properties of logarithms to any base.

25. Solve equations involving logarithms to any base.

26. Explain the development of the common logarithm.

27. Use a table of.mantissas to find the common logar-ithm of a real number.

28. Use a table of mantissas to find antilogarithms.

29. Use the method of linear interpolation to approximatemantissas and antilogarithms not found in a table.

30. Organize and use common logarithms to perform theoperations of multiplication, division, raising topowers, and extracting roots.

31. Solve exponential equations using common logarithms.

M 94 p. 1

LEVEL OBJECTIVE

THE STV)ENT WILL APPLY SUCH CONCEPTS AS COMPLEX NUMBERS,MATHEMA.ICAL INDUCTION, PROBABILITY, AND TRIGONOMETRY TOPROBLEM SOLVING SITUATIONS BY COMPLETING THE FOLLOWINGLEVEL TO THE SATISFACTION OF THE TEACHER.

I. Complex Numbers

Objective #1: The student will apply the concept of com-plex numbers to problem solving by com-pleting the following suggested activitiesto the satisfaction of the teacher.

Activities:


1. Define a complex number.

2. Express any number in complex form.

3. State the sequence of the powers of "i."

4. Perform the operations of addition, subtraction,multiplication, and division of complex numbers.

5. Graph complex numbers on a coordinate plane.

6. List the properties of the complex field.

II. Sequences, Series, and Mathematical Induction

Objective #2: The student will apply the properties ofprobability to solving problems aboutuncertainity and chance by completing thefollowing suggested activities to thesatisfaction of the teacher.

Activities:


7. Identify an arithmetic sequence of numbers.

8. Find the nth term and the sum of the first n termsin an arithmetic series.

M 94 p. 2

9. Identify a geometric sequence of numbers.

10. Find the nth term and the sum of the first n termsin a geometric series.

11. Solve verbal problems involving arithmetic andgeometric sequences.

12. Insert arithmetic and geometric means between twoextremes.

13. Solve verbal problems involving arithmetic andgeometric series.

14. State the axiom of mathematical induction.

15. Use mathematical induction in proofs.

III. Permutations, Combinations, and Probability

Objective #3: The student will apply the properties ofprobability to solving problems about un-certainity and chance by completing thefollowing suggested activities to thesatisfaction of the teacher.

Activities:


16. Define a permutation.

17. Solve verbal proble 'ns involving peromtations.

18. Define a combination or a selection.

19. Distinguish between a permutation and a combinationtype problem.

20. Solve verbal problems involving combinations.

21. Identify the basic probability situations.

22. Solve verbal problems involving probabilities.

M 94 p. 3

IV. Trigonometry

Objective #4: The student will apply trigonometricfunctions to problem solving situationsby completing the following suggestedactivities to the satisfaction of theteacher.

Activities:


23. Define an angle in standard position.

24. Define the six trigonometric functions.

25. Recognize the relationships that exis+ among thetrigonometric functions.

26. Manipulate the fundamental identities.

27. Find the trigonometric functions of special angles.

28. Find the trigonometric functions of any angle usingthe tables of values.

29. Interpolate the values of the trigonometric func-tions.

30. Solve right triangles.

31. To solve right triangles in verbal problem situa-tions.

M 95 p. 1

LEVEL OBJECTIVE

THE STUDENT WILL APPLY THE PRINCIPLES OF GEOMETRY BY COM-PLETING THE FOLLOWING LEVEL TO THE SATISFACTION OF THETEACHER.

I. Common Sense and Organized Knowledge

Objective #1: The student will increase in knowledge ofthe origin and definitions of geometry bycompleting the following suggested activ-ities to the satisfaction of the teacher.

Activities:


1. Trace the early histroy of Geometry from the Greeksand Egyptians.

2. Recognize the common sense used by the Egyptians.

3. Recognize the logical reasoning used by the Greeks.

4. Draw geometric abstractions.

5. Define a theorem as a sta;;ement which can be proved.

6. Define postulates and axioms as statements whichare accepted without proof.

7. Accept point, line, and plane as the basic undefinedterms in Euclidian Geometry.

3raw representations for the basic undefined terms.

II. Sets and Real Numbers

Objective #2: The student will increase in comprehen-sion of sets and the real number set inparticular by completing the followingsuggested activities to the satisfactionof the teacher.

Activities:


M 95 p. 2

9. Define and use the following terminology:

a. set,

b. "is an element of,"

c. subset,

d. empty set,

e. intersection of two sets, and

f. union of two sets.

10. Construct the subsets of a given set.

11. Determine the intersection of two sets.

12. Determine the union of two sets.

13. Establish one-to-one correspondences between sets.

14. Recognize the structure of the real number system.

15. Use the notations and I .

16. State the following basic properties of the realnumber system:

a. Uniqueness of order,

b. Transitivity of order,

c. Addition for inequalities, and

d. Multiplication for inequalities.

17. Use absolute value notation.

18. Compute absolute value.

III. Distance

Objective #3: The student will increase in comprehen-sion of length and distance by complet-ing the following suggested activitiesto the satisfaction of the teacher.

M 95 P. 3

Activities:


19. Define distance as the unique positive number whichcorresponds to each pair of distant points.

20. Apply the Ruler Postulate to the computation of dis-tance.

21. Set up a coordinate system for a line.

22. Define point B lying between points A and C.

23. Understand the introductory theorems concerning be-tweeness of points.

24. Define basic terms in geometry such as:

a. segment,

b. endpoints,

c. length of a segment,

d. ray,

e. interior of a ray, and

f. opposite. rays.

25. Represent segments, rays, and lines by appropriatenotation.

26. Represent segments, rays and lines by appropriatediagrams.

IV. Lines, Planes, and Separation

Objective #4: The student will increase in comprehen-sion of lines, planes, and space by com-pleting the following suggested activ-ities to the satisfaction of the teacher.

Activities:


1

[

M 95 p. 4

27. Recognize space as the set of all points.

28. Distinguish between collinear and noncollinearpoints.

29. Distinguish between coplanar and noncoplanarpoints.

30. Recognize that a plane contains at least three non-collinear points.

31. Recogn:gLe that space contains at least four non-coplanar points.

32. Determine minimum conditions for a plane.

33. Draw diagrams as representations of geometric ab-stractions.

34. Determine the hypothesis and conclusion of a con-ditional statement.

35. Define a convex set of points.

36. Explain how a line separates a plane into two non-empty convex sets.

37. Explain how a plane separates space into two non-empty, concux sets.

V. Angles and Triangles

Objective #5: The student will apply the properties ofangles, triangles, and congruence to solv-ing problems by completing the followingsuggested activities to the satisfactionof the teacher.

Activities:


38. Define an angle as the union of two noncollinearrays that have a common endpoint.

39. Identify ',ne vertex and sides of the angle.

M 95 p. 5

40. Recognize an angle as a plane figure.

41. Distinguish between the interior and exterior of anangle.

42. Define a triangle as the union of the three seg-ments which join, in order, three noncollinearpoints.

43. Identify the vertices, sides, and angles of a tri-angle.

44. Identify included sides and included angles.

45. Identify opposite angles.

46. Recognize a triangle as a plane figure.

47. Distinguish between the interior and exterior of atriangle.

48. Define the measure of an angle as the number of de-grees it contains.

49. Identify a linear pair of angles.

50. Identify adjacent angles as two angles in a planethat have a common side but no interior points incommon.

51. Identify supplementary angles as two angles the sumof whose measures is 180 degrees.

52. Define a right anbles as an angle whose measure is90 degrees.

53. Define perpendicular sets as line, rays, or seg-ments which intersect to form right angles.

54. Identify complementary angles as two angles the sumof whose measures is 90 degrees.

55. Define an acute angle as an angle whose measure isless than 90 degrees.

56. Define an obtuse angle as an angle whose measure isgreater than 90 degrees.

M 95 p. 6

57. Recognize congruent angles as angles that have thesame measure.

58. Define vertical angles as two angles whose union isof two intersecting lines.

59. Define congruent segments as segments which havethe same length.

60. Recognize the reflexive, symmetric, and transitiveproperties of a relation.

61. Identify an equivalence relation as a relation whichpossesses the reflexive, symmetric, and transitiveproperties.

62. Identify corresponding sides and angles of triangles.

63. Define a congruence between triangles as a corre-spondence in which corresponding sides and corre-sponding angles are congruent.

64. Recognize the following minimum conditions for de-termining congruence between triangles:

a. SAS (Side-Angle-Side)

b. ASA (Amgle-Side-Angle)

c. SSS (Side-Side-Side)

65. Apply the basic congruence situations in the proofsof original exercises.

4

66. Prove and apply the Isosceles Triangle theorem.

67. Distinguish among isosceles, equilateral, and scalenetriangles.

68. Define a median of a triangle as a segment with oneendpoint at a vertex of the triangle and the otherendpoint at the midpoint of the opposite side.

69. Define an angle bisector as a segment whose end-points are the points of intersection of the trian-gle and a ray that bisects an angle of the triangle.

M 95 p. 7

VI. A Closer Look At Proof

Objective #6: The student will apply the postulates,axioms, and theorems to proofs of mathe-matical statements by completing the fol-lowing suggested activities to the satis-faction of the teacher.

Activities:


70. Examine and discuss the nature of indirect proof.

71. Use indirect proof in original exercises.

72. Distinguish between existence (at least one) anduniqueness (at most one).

73. Examine proofs which involve both existence anduniqueness properties.

74. Prove and apply the theorem which states that givena plane, a line in the plane, and a point on theline, there is exactly one line that is in the givenplane, contains the given point, and is perpendicu-lar to the given line.

75. Define the perpendicular bisector of a segment in aplane as the line in the plane that is perpendicularto the segment at its midpoint.

76. Prove and apply the theorem which identifies theperpendicular bisector of a segment in a plane asthe set of all points in the plane that are equid-istant from the endpoints of the segment.

77. Prove and apply the theorem which states that givena line and point not on the line, there is at mostone line that contains the given point and is per-pendicular to the given line.

78. Define a right triangle as a triangle that has oneright angle.

79. Identify the hypotenuse and legs of a right triangle.

M 95 P. 8

80. Prove and apply the theorem which states that givena line and a point not on it, there is at least oneline that contains the given point and is perpendic-ular to the given line.

81. Introduce auxiliary sets when the need arises.

82. Prove and apply the ASA and SSS congruence theorems.

VII. Geometric Inequalities

Objective #7: The student will apply the properties ofgeometric inequalities to proofs of mathe-matical statements by completing the fol-lowing suggested activities to the satis-faction of the teacher.

Activities:


83. Define an exterior angle of a triangle as an anglethat forms a linear pa7 with one of the angles ofthe triangle.

84. Define remote interior angles as those angles of a

triangle with which the exterior angle does not forma linear pair.

85. Prove and apply the theorem which states that themeasure of an exterior angle is greater than themeasure of each remote interior angle.

86. Prove and apply the SAA case of congruence.

87. Prove and a'ply the H.L. case of congruence in aright triangle.

88. Prove and apply the theorem which states that iftwo sides of a triangle are not congruent, the an-gles opposite them are not congruent, and the larg-er angle is opposite the longer side.

89. Prove and apply the theorem which states that iftwo angles of a triangle are not congruent, thenthe sides opposite them are not congruent, and thelonger side is opposite the larger angle.

M 95 p. 9

90. Prove and apply the theorem that the sum of thelengths of any two sides of a triangle is greaterthan the length of the third side.

91. Prove and apply the theorem which states that theshortest segment joining a point to a line is theperpendicular segment.

92. Define the distance between a line and a point noton the line as the length of the perpendicular seg-ment from the point to the line.

93. Define an altitude of a triangle as the perpendicu-lar segment from any vertex to the line that con-tains the opposite side.

94. Prove and apply the theorem which states that if twosides of one triangle are congruent respectively totwo sides of a second triangle, and if the measureof the included angle of the first triangle is great-er than the measure of the included angle of the sec-ond, then the third side of the first triangle islonger than the third side of the second.

95. Prove and apply the theorem which states that if twosides of one triangle are congruent respectively totwo sides of a second triangle, and if the third sideof the first triangle is longer than the third sideof the second, then the included angle of the firsttriangle is larger than the included angle of thesecond.

VIII. Perpendicular and Parallel Lines and Planes in Space

Objective #8: The student will apply axioms and theo-rems to proofs involving paralles andperpendicular lines and planes in spaceby completing the following suggestedactivities to the satisfaction of theteacher.

Activities:


96. Identify a line and a plane as being perpendicularif (1) they intersect and (2) the given Tine isperpendicular to every line that is in the planeand intersects the given line.

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97. Identify a segment or ray as being perpendicular toa plane if (1) it intersects the plane, and (2)the line that contains it is perpendicular to theplane.

98. Prove and apply the theorem which states that iftwo points, P and Q are each equidistant from twopoints A and B, then every point between P and Qis equidistant from A and B.

99. Prove and apply the theorem which states that ifa line is perpendicular to each of two intersect-ing lines at their point of intersection, then itis perpendicular to the plane containing them.

100. Prove and apply the theorem which states that forany line and any point of the line there exists a

plane perpendicular to the line at the point.

101. Prove and apply the theorem which states that forany plane and any point of the plane there existsa line perpendicular to the plane at the point.

102. Prove and apply the theorem which states that ifa line and a plane are perpendicular, then theplane contains every line perpendicular to thegiven line at its point of intersection with theplane.

103. Prove and apply the theorem which states that forany line and'any point of the line there is at mostone plane perpendicular to the line at the point.

..

104. Define the perpendicular bisecting plane of a seg-ment as the plane perpendicular to the segment atits midpoint.

105. Prove ane apply the theorem which states that forany plane and any point of the plane there is atmost one line perpendicular to the plane at thepoint.

106. Prove and apply the theorem which states that theperpendicular bisecting plane of a segment is theset of all points equidistant from the endpointsof the segment.

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107. Prove and apply the theorem which states that twolines perpendicular to the same plane are coplanar.

108. Prove and apply the theorem which states that forany point and any line there is exactly one planecontaining the given point and perpendicular to thegiven 14ne.

109. Prove and apply the theorem which states that forany point and any plane there is exactly one linecontaining the given point and perpendicular to thegiven plane.

110. Define the distance to a plane from a point not onthe plane as the length of the perpendicular segmentfrom the point to the plane.

111. Prove and apply the theorem which states that theperpendicular segment to a plane from a point noton the plane is shorter than any other segment fromthe point to the plane.

112. Define two lines as being parallel if they are co-planar and do not intersect.

113. Define two lines as being skew if they are not co-planar and do not intersect.

114. Prove and apply the theorem stating that two par-allel lines lie in exactly one plane.

115. Prove and apply the theorem stating that if twolines in the same plane are both perpendicular tothe same line, then they are parallel.

116. Prove and apply the existence theorem about paral-lelism.

117. Define and identify transversals, alternate-inter-ior angles, corresponding angles, interior angleson the same side of the transversal, and alternate-exterior angles.

118. Prove and apply those theorems which establish thattwo lines are parallel.

119. Define the Parallel Postulate.

M 95 p. 12

120. Apply the Parallel Postulate to prove other theorems.

121. Prove and apply those theorems stating that if twoparallel lines are cut by a transversal, then al-ternate interior angles are congruent, each pair ofcorresponding angles are congruent, and any two in-terior angles are supplementary if their interiorsintersect.

122. Prove and apply those theorems stating that in aplane, two lines parallel to the same line are par-allel to each other and that if a line is perpendic-ular to one of two parallel lines it is perpendicu-lar to the .her.

123. Relate par ,ism to triangles and quadrilaterals.

124. Define a quadrilateral.

125. Classify quadrilaterals and identify the propertiesof each.

126. Prove and apply the statement that the segment join-ing the midpoints of two sides of a triangle is par-allel to the third side, and the length of the seg-ment is half the length of the third side.

127. Prove and apply those theorems concerned with trans-versals to several parallel lines.

128. Define parallel planes.

129. Prove and apply the theorems concerned with paral-lel planes.

130. Prove and apply the theorems concerned with paral-lels and perpendiculars.

131. Define a dihedral angle.

132. Define and apply all properties of a dihedral angle.

IX. Areas of Polygonal Regions

Objective #9: The student will apply methods of findingareas to a variety of polygonal regions bycompleting the following suggested activ-ities to the satisfaction of the teacher.

M 95 p. 13

Activities:


133. Define a function.

134. Define a function from A into B.

135. Define triangular region.

136. Define a polygonal region.

137. Apply the Area Postulate.

138. Apply the congruence Postulate for Area.

139. Apply the Area Addition Postulate.

140. Apply the Unit Area Postulate.

141. Develop and compute areas of polygonal regions.

142. Prove and apply the Pythagorean Theorem.

143. Prove and apply those theorems concerning the 30-60right triangle.

X. Similarity

Objective #10: The student will apply properties of sim-ilarity to proofs of statements involvingsimilar figures by completing the follow-ing suggested activities to the satisfac-tion of the teacher.

Activities:


144. Derine a sequence, an infinite sequence and a finitesequence.

145. Distinguish among different types of sem.ences.

146. Define two sequences which are proportional.

147. Define the constant of proportionality.

M 95 p. 14

148. Prove and apply the theorem stating that propor-tionality of sequences is an equivalence relation.

149. Define similarity or triangles.

150. Apply the algebraic properties of a simple propor-tion.

151. Prove and apply the basic proportionality theorem.

152. Prove and apply the theorem which states that if a

line intersects two sides of a triangle and cutsoff corresponding segments that are proportional tothe two sides, then the line is parallel to thethird side.

153. Prove and apply the AAA similarity theorem.

154. Prove and apply the SAS similarity theorem.

155. Prove and apply the SSS similarity theorem.

156. Prove and apply the theorem which states that forany given right triangle, the three triangles con-tained in the union of the given triangle and thealtitude to the hypotenuse are similar.

157. Prove and apply the theorem which states that if twotriangular regions are similar, then the ratio oftheir areas is the square of the ratio of the lengthsof any two corresponding sides.

158. Prove and apply the theorem which states that if 3parallel lines are cut by two transversals, thenthe lengths of the segments ir ercepted on onetransversal are proportional to the lengths of thecorresponding segments intercepted on the othertransversal.

XI. Circles and Spheres

Objective #11: The student will apply the properties ofcircles and spheres to proofs and prob-lems involving circles and spheres bycompleting the followihg suggested ac-tivities to the satisfaction cf the tea-cher.

M 95 p. 15

Activities:


159. Define a sphere with center P and radius R as theset of all points at a distance R from P.

160. Define a circle with center P and radius R as theset of all points in a plane at a distance R fromP.

161. Define the diameter of a circle as the number thatis twice the radius.

162. Prove and apply the theorem which states that theintersection of a sphere and any plane containingthe center of the sphere is a circle with the samecenter and radius as the sphere.

163. Define the following terms related to circles andspheres:

a. great circle,

b. chord,

c. secant,

d. diametral chord,

e. radial segment,

f. tangent line,

g. tangent plane, and

h. point of tangency.

164. Prove and apply the theorem concerning the interiorpoints of a circle.

165. Prove and apply the theorem which states that aline in the plane of a circle is tangent to thecircle if and only if it is perpendicular to a

radial segment.

M 95 p. 16

166. Prove and apply the theorem which states that thesegment joining the center of a circle to a pointof the chord bisects the chord if and only if thesegment is perpendicular to the chord.

167. Define the distance from the center of a circle(or sphere) to a chord as the distance between thecenter and the line containing the chord.

168. Prove and apply the theorem which states that fora circle or for two circles of equal radii, twochords are equidistant from the center if and onlyif the chords are congruent.

169. Prove and apply the theorems concerned with the in-tersection of planes with a sphere.

170. Define the following terms related to circles:

a. central angle,

b. minor arc,

c. major arc, and

d. semicircle.

171. Find the degree measures of both major and minorarcs.

172. Prove and apply the arc addition theorem.

173. Define intercepted arcs and inscribed angles.

:74. Prove and apply the theorem stating that the measureof an inscribed angle is half the measure of the in-tercepted arc.

175. Prove and apply the theorem stating that in a cir-cle or in congruent circles, if two chords are con-gruent, then their corresponding arcs are congru-ent.

176. Prove and apply the converse of the above theorem.

E

M 95 p. 17

177. Prove and apply the theorems concerned with themeasures of various angles in a circle.

178. Prove and apply the Two Circle theorem.

179. Define the following terms related to circles:

a. common tangent,

b. external common tangent,

c. internal common tangent,

d. circles tangent internally, and

e. circles tangent externally.

180. Prove and apply the theorems concerning tangent andsecant segments in a circle.

181. Define a polygon and the following related terms:

a. vertices,

b. sides,

c. diagonal,

d. angles, and

e. perimeter.

182. Define a regular polygon as one which (a) is pseudo-convex (b) has congruent sides, and (c) has con-gruent angles.

183. Define an inscribed polygon.

184. Define the area of the region bounded by a regularn-gon is 1/2 aen, where a is the apothem and e isthe length of each side.

185. Prove and apply the theorem stating that the ratio,C, of the circumference to the diameter is the samefor all circles.

M 95 p. 18

186. Prove and apply the theorem stating that the areaof a circular region of radius R is given by R2.

187. Define the length of arc A as r m (AB).180

188. Define the area of sector AB of radius r as mAB360

r 2.

XII. Concurrency and Constructions

Objective #12: The student will apply properties of :on-struction by completing the followingsuggested activities to the satisfactionof the teacher.

Activities:

The student will he able to:

189. Given an angle and a ray, construct an angle, withthe given ray as a side, congruent to the given an-gle.

190. Given an angle, construct the ray that bisects theangle.

191. Given a point on a line in a plane, construct aline perpendicular to the given line at the givenpoint.

192. Given a point outside a line, construct a perpendic-ular to the line from the point.

193. Given a segment, construct the perpendicular bisec-tor of the segment.

194. Given a point outside a line, construct a line par-allel to the given line through the given point.

195. Given an arc of a circle, construct a line thatbisects the arc.

196. Given a point on a circle, construct the tangent tothe circle at the given point.

M 95 p. 19

197. Given a point outside a circle, construct the tan-gents to the circle from the point.

198. Given a triangle, circumscribe a circle about thetriangle.

199. Given a triangle, inscribe a circle in the tri-angle.

M 96 p. 1

LEVEL OBJECTIVE

THE STUDENT WILL APPLY THE PRINCIPLES OF MATHEMATICS TOPRACTICAL PROBLEMS BY COMPLETING THE FOLLOWING LEVEL TO THESATISFACTION OF THE TEACHER.

I. Numbers

Objective #1: The student will increase in comprehen-sion of numbers by completing the fol-lowing suggested activities to the sat-isfaction of the teacher.

Activities:


1. Express quantity by the use of numbers.

2. Write the integers which are whole numbers.

3. Tell which digits are used to indicate a total num-ber.

4. Determine the difference between face value andplace value when working with different numera-tions.

5. Write the place values of the digits for any num-ber using base 10.

6. Write the place values and numerical value of eachdigit in a number using base 10.

7. Write the numerical value of a number written inbase five.

8. Express a number using words.

9. Round off a number to nearest ten, hundred, thou-sand, etc.

10. Round off an answer when dealing with money to thenearest dollar.

11. Read a decimal fraction.

M 96 p. 2

12. Express common fractions as decimal fractions.

13. Round off decimal fractions to any particular placevalue.

14. Express per cent as a decimal fraction.

15. Express decimal fractions as per cent.

16. Express common fractions as per cent.

17. Use a bar graph to determine a title, what is shownon the horizontal axis and vertical axis, and re-late the picture to data given.

18. Make a bar graph.

19. Read a broken-line graph.

20. Construct a broken-line graph.

21. Read a picture graph.

22. Read a circle or rectangular graph.

23. Construct a circle or rectangular graph.

II. Statistics

Objective #2: The student will increase in comprehen-sion of statistics by completing the fol-lowing suggested activities to the sat-isfaction of the teacher.

Activities:


24. Find the arithmetic average of a set of numbers.

25. Illustrate the distributive law with respect to theproduct of two numbers.

26. Set up a frequency table with columns of values,tally marks frequency, and sum.

M 96 p. 3

27. Determine the median of a set of numbers by arrang-

ing them in order of size.

28. Determine the mode of a set of numbers.

29. Determine tie mean and medium from grouped data.

30. Determine the mid-point of grouped data.

III. Application of Numbers

Objective #3: The student will apply the properties andoperations of numbers by completing thefollowing suggested activities to thesatisfaction of the teacher.

Activities:


31. Add numbers horizontally by making use of the com-

mutative law for addition.

32. Add numbers vertically and horizontally making use

of the associative law for addition.

33. Estimate the product of two numbers and then com-

pute it.

34. Compute the per cent of a number.

35. Apply the finding of per cent to the weekly income

tax of a weekly salary.

36. Illustrate the meaning of a fractional per cent as

a fractional part of one per cent.

37. Subtract numbers in a horizontal arrangement.

38. Set up a payroll roster and apply horizontal addi-

tion and subtraction to determine net pay.

39. Fill out a currency break-down containing the ex-act number and kinds of bills and coins needed to

meet a particular payroll.

IV. Measurements

Objective #4: The student will increase in knowledge of

M 96 p. 4

measurements by completing the followingsuggested activities to the satisfactionof the teacher.

Activities:


40. Measure an object by comparing it with some unitof measure.

41. Use a ruler to measure an object.

42. Illustrate that all measurements are approximateand precision of a measurement depends upon thesize of the smallest unit with which the measure-ment is made.

43. Change a mixed number to an improper fraction.

44. Change an improper fraction to a mixed number.

45. Determine the maximum error in any measure.

46. Change a fraction to an equivalent fraction.

47. Find the prime factors of a number.

48. Find the common factors of two numbers.

49. Simplify a fraction as far as possible.

50. Add two proper fractions.

51. A,,1 a proper fraction and a mixed number.

52. Add a mixed number to a mixed number.

53. Subtract two fractions or mixed numbers.

54. Multiply two fractions and a mixed number.

55. Multiply a proper fraction and a mixed number..

56. Multiply two mixed numbers.

57. Divide two proper fractions.

M 96 P. 5

58. Divide a fraction by a mixed number.

59. Divide a mixed number by a mixed number.

60. Determine the accuracy and significant digits in anumber.

61. Add two decimal fractions.

62. Subtract two decimal fractions.

53. Compute upper and lower limit of a measure whengiven the tolerance.

64. Multiply two decimal fractions.

65. Change a common fraction to a decimal fraction.

66. Change decimal fractions to common fractions.

67. Divide a decimal fraction by a whole number.

68. Divide a decimal fraction by a decimal fraction.

69. Illustrate that the metric system is a decimal sys-tem.

70. Change metric values to English equivalents.

V. Geometry

Objective #5: The student will apply geometrical con-cepts to problems of geometry iv complet-ing the following suggested activities tothe satisfaction of the teacher.

Activities:


71. Describe and identify a line, plane and point.

72. Define and name a line segment.

73. Define and name an angle and its parts.

M 96 p. 6

74. Define a straight angle, right angle, obtuse an-gle, acute angle, and a reflex angle.

75. Measure an angle with the use of a protractor.

76. Define perpendicular lines, parallel lines, and thedistance between two parallel lines.

77. Define corresponding angles and alternate interiorangles.

78. Copy a given angle with a straight edge and com-pass.

79. Construct a line parallel to a given line throughan outside point.

80. Divide a segment into any number of equal segmentsby using parallel lines.

81. Define congruent triangles.

82. Define corresponding parts of congruent triangles.

83. Define triangles classified according to angles(acute, obtuse, right).

84. Define triangles classified according to sides(scalene, isosciles, and equilateral).

85. Prove triangles congruent by using the S.A.S.theorem.

86. Define vertical angles.

87. Prove triangles congruent by using the A.S.A.theorem.

88. Prove triangles congruent by using the S.S.S.theorem.

89. Define the ratio of two numbers.

90. Apply ratio to the measures of sides of atriangle or triangles.

91. Define a proportion.

92. State the properties of a proporption involvingmeans and extremes of the proportion.

M 96 P. 7

93. Use a proportion for solving problems in which ;:er-tain kinds of comparisms are made.

94. Define similar triangles.

95. Prove triangles similar by using the AA theorem.

96. Apply the proportions developed from similar tri-angles to find particular sides of heights of ob-jects.

97. Define inductive reasoning.

98. Apply inductive reasoning to determine the sum ofthe measures of the angles of a triangle, an exter-ior angle of a triangle, and the sum of the measuresof the angles of a quadrilateral.

99. Define a parallelogram and state its character-istics.

100. Defind deductive reasoning.

101. Test the validity of an argument by using diagramsand deductions.

102. Prove corresponding parts of congruent trianglescongruent.

VI. Formulas Dealing with Perimeter, Area and Volume

Objective #6: The student will apply geometrical conceptsto problems of perimeter, area, and volumeby completing the following suggestedactivities to the satisfaction of the tea-cher.

Activities:


103. Define a formula.

104. Find the perimeter of any polygon.

105. Define the radius of a circle.

106. Find the circumference of a circle.

107. Define/77 (pi).

M 96

108. Find the area of a rectangle.

109. Find the area of a parallelogram.

110. Find the area of a triangle.

111. Find the area of a trapezoid.

112. Find the area of a circle.

113. Find the volume of a rectangular solid.

114. Find the volume of a cylinder.

115. Find the volume of a cone.

116. Find the volume of a sphere.

VII. Algebra

Objective #7: The student will apply abstract reasoningto algebraic properties by completingthe following suggested activities tothe satisfaction of the teacher.

Activities:


117. State in words ':e meaning of particularalgebraic expre.: ions.

118. Illustrate the commutative law for additionand multiplication.

119. Translate a word statement into an algebraicexpression.

120. Define a variable.

121. Find the value of a numerical expression.

122. Find the relaticn eAisting between two variables.

123. Define a statement or equation.

124. Solve an equation in one variable.

M 96 p. 9

125. Solve an equation by division.

126. Solve an equation by suing multiplication.

127. Solve an equation by suing subtraction.

128. Solve an equation by using addition.

129. Solve an wquation requiring more than one operation.

130. Combine like terms in an equation.

131. Solve verbal problems which require the use of equa-tions.

VIII. Signed Numbers

Objective #8: The student will apply the properties ofnumbers to include signed numbers by com-pleting the following suggested activitiesto the satisfaction of the teacher.

Activities:


132. Define a signed number.

133. Represent changes by using signed numbers.

134. Illustrate a number scale.

135. Add signed numbers.

136. Apply the commutative low of addition to signed num-bers.

137. Find the absolute value of a number, signed or un-signed.

138. Subtract signed numbers.

139. Multiply signed numbers.

140. Combine like terms involving signed numbers.

141. Apply the distributive law to expres:ions with pa-rentheses.

M 96 p. 10

142. Solve equations involving signed numbers.

143. Graph an equation which is a function.

144. Read a graph which represents an equation.

145. Graph equations which involve signed numbers.

146. Read order pairs from a graph of an equation in-volving signed numbers.

IX. Application of Formulas

Objective #9: The student will apply formulas to problemsolving situations by completing the follow-ing suggested activities to the satisfactionof the teacher.

Activities:


147. Apply known formulas and given formulas to determinethe solution of a 'physical problem.

148. Name the sides of a right triangle.

149. State the relation between the acute angles of aright triangle.

150. State the Pythagorean Theorem.

151. Find the missing side of a right triangle when giventhe measure of the other two sides.

152. Find the perfect square of a number.

153. Find the square root of a number from a table ofsquares.

154. Find the square root by using division.

155. Determine the actual length of an object by usinga given scale.

156. Change the actual measurements of an object to thoseof a mailer represnetation.

M 96 p. 11

157. Find the perimeter and area of an object of a scaledrawing.

158. Find the actual distance between two points on amap.

M 97 p. 1

LEVEL OBJECTIVE

THE STUDENT WILL APPLY THE PRINCIPLES OF NUMBERS AND POLY-NOMIALS BY COMPLETING THE FOLLOWING LEVEL TO THE SATISFAC-TION OF THE TEACHER.

I. Numbers and Sets

Objective #1: The student will apply the principles ofnumbers and sets by completing the fol-lowing suggested activities to the sat-isfaction of the teacher.

Activities:


1. Determine the elements of a set.

2. Represent a set using the roster method.

3. Represent a set using the rule method.

4. Find subsets of a set.

5. Recognize the set of no elements as the null set.

6. Recognize a finite set.

7. Recognize an infinite set.

8. Recognize equal sets.

9. Recognize equivalent sets.

10. Set up a one to one correspondence between the ele-ments of equivalent sets.

11. Identify the set of counting numbers.

12. Determine thesuccessor of any whole number.

13. Determine multiples of a number.

14. Identify odd or even numbers.

M 97 p. 2

15. Use the number line for indentifying the relations"is equal to," "is not equal to," "is greater than,"and "is less than."

16. Identify the rational numbers of arithmetic.

17. Distinguish between a numeral and a number.

18. Simplify numerical phrases.

19. Follow the order of operations for ungrouped orgrouped numerical phrases.

20. Use grouping symbols in numerical phrases.

21. Use symbols and numerals to form sentences,

22. Determine that a sentence is true or false.

23. Recognize a variable as a symbol that may be re-placed by any element of a specified set.

24. Recognize the domain of the variable.

25. Determine the value of an open phrase.

26. Represent word phrases by open phrases.

27. Use variables to represent number patterns.

28. Determine the truth number for an open sentence.

29. Determine the truth set of an open sentence.

30. Use exponents to abbreviate phrases involving

a repeated base.

31. Graph truth sets on the number line.

32. Determine truth values for compound sentencesusing "and" and "or" as connectives.

33. Determine the intersection of sets.

34. Graph compound sentences whose connectives are"and" or "on"

M 97 p. 3

35. Use the addition and multiplication properties ofzero.

36. Use the multiplication property of one.

37. Recognize that the identity elements for additionand multiplication are zero and one.

38. Recognize binary operations.

39. Recognize a closure property.

40. Recognize and use the commutative property foraddition.

41. Recognize and use the associative property foraddition.

42. Recognize and use the commutative property formultiplication.

43. Recognize and use the associative property formultiplication.

44. Recognize and use the distributive law for mul-tiplication over addition.

II. Operations and Polynomials

Objective #2: The student will apply properties andoperations of the real number systemto algebraic expressions by complet-ing the following suggested activitiesto the satisfaction of the teacher.

Activities:


45. Simplify phrases by using properties of numbers.

46. Find the product of a monomial and a monomial.

47. Find the product of a monomial and a binomial orpolynomial.

48. Find the product of two binomials.

49. Find factors of a number.

50. Test divisibility by 2, 3, 4, 5.

M 97

51. Identify a prime number.

52. Identify a composite number.

53. Factor a number into a product of primes.

54. Recognize that prime factor is unique.

55. Apply exponents as an abbreviation for repeat-ed factors.

56. Use fractions to name numbers.

57. Multiply fractions.

58. Find the least common multiple of two or mor.:counting numbers.

59. Add fractions.

60. Find the reciprocal of a number.

61. State that a ratio is the comparison of twonumbers by division.

62. Write a proportion.

63. Determine a true proportion.

64. Determine equivalent proportions.

65. Apply proportions in solving problems.

66. Convert fractions to per cent.

67. Apply per cent in solving problems.

68. Identify units of measure.

69. Determine greatest possible error.

70. Determine precision of a measurement.

71. Convert among standard units of measure.

72. Find area of certain-plane figures.

73. Define square units of measure.

p. 4

M 97 p. 5

74. Find the volume of certain solid figures.

75. Define cubic units of measure.

76. Find perimeters of certain plane figures.

77. Convert word phrases to open phrases.

78. Form open sentences from word sentences.

79. Solve number problems.

80. Solve problems involving perimeters and areas.

81. Solve problems in costs.

82. Solve distance problems.

83. Classify subsets of numbers of the real numbers.

84. Perform operations of addition, subtraction, mul-tiplication, and division with the set of real num-bers.

85. Recognize order in the set of real numbers.

86. Apply the concepts of absolute value with respectto distances on number lines.

87. Apply properties of addition and multiplication de-veloped on counting numbers to the set of real num-bers.

88. Determine equivalent open sentences.

89. Use the addition property of equality to solve equa-tions.

90. Use the multiplication property of equality to solveequations.

91. Add polynomials.

92. Multiply polynomials.

93. Subtract polynomials.

M 97


95. Determine the probability of a favorable outcomefrom given conditions of uncertainty.

p. 6

96. Recognize the difference between experimental prob-ability and mathematical probability.

M 98 p. 1

LEVEL OBJECTIVE

THE STUDENT WILL APPLY ABSTRACT REASONING TO THE REAL NUMBERSYSTEM AND ALGEBRAIC EXPRESSIONS BY COMPLETING THE FOLLOWINGLEVEL TO THE SATISFACTION OF THE TEACHER.

I. Real Numbers and Order

Objective #1: The student will increase in knowledge oforder and the principles of real numbersby completing the following suggested ac-tivities to the satisfaction of the tea-cher.

Activities:


1. Use the symbol for the opposite of a number.

2. Write the absolute value of a number.

3. Add, subtract, multiply, and divide:

a. two positive numbers,

b. two negative numbers, and

c. one positive and one negative number

4. Recognize and identify the following properties ofthe real numbers:

a. closure,

b. identity,

c. inverse,

d. commutative,

e. associative, and

f. distributive.

5. Recognize and combine like terms.

M 98 D. 2

6. Use addition and multiplication properties to solvesimple open sentences.

7. Solve simple verbal problems.

8. Solve and graph simple inequalities.

9. Recognize and identify the following properties oforder:

a. comparison property,/

b. order property of oppositbs,

c. transitive,

d. addition, and

e. multiplication.

10. Solve and graph inequalities using the above proper-ties.

11. Solve and graph compound inequalities.

II. Powers and Roots

Objective #2: The student will increase in knowledge ofpowers and roots by completing the follow-ing suggested activities to the satisfac-tion of the teacher.

Activities:

The studeJt will be able to:

12. Identify and differentiate between exponent, cof-ficient, base and power.

13. Apply order of operations to simplify expressions.

14. Apply the following properties of exponentF:

a. xa x5 = xa+5

b. xa5

= xa-5 where x # 0,7(

1

c. x-a - 7(13 where x # 0,

M 98 p. 3

d. x° = 1 where x # 0,

e. (xy)n = xn yn,

f. (x)n . xnwhere y # 0,(y) .7

g. (xa)b = xab.

15. write a real number in scientific notation.

16. Find the square root of a number.

17. Find the square of a number.

18. Recognize perfect squares.

19. Determine the prime factorization of a number.

20. Differentiate between rational and irrational num-bers.

21. Multiply numbers in radical form.

22. Simplify radical numbers and expressions.

23. Combine radical numbers and expressions.

24. Divide numbers in radical form.

25. Rationalize the denominator.

26. Approximate square root by division.

III. Polynomials

Objective #3: The student will increase in knowledge ofoperations anr1 properties of polynomialsby completing the following suggested ac-tivities to the satisfaction of the tea-cher.

Activities:


27. Recognize and differentiate between monomial, bi-nomial, trinomial, and polynomial.

M 98 P. 11'

28. Identify the coefficient of a monomial.

29. Identify the degree of a polynomial.

30. Identify the factor common monomial factors.

31. Factor quadratic trinomials.

32. Factor quadratic polynomials.

33. Identify and factor the binomial called differenceof two squares.

34. Identify and factor perfect trinomial squares.

35. Factor polynomials completely.

IV. Relations and Functions

Objective #4: The student will increase- in knowledge ofrelations, functions, and their graphs bycompleting the following suggested activ-ities to the satisfaction of the teacher.

Activities:


36. Recognize the following terms:

a. coordinate of a point,

b. ordered pair,

c. x-axis,

d. y-axis,

e. origin,

f. coordinate plane, and

g. quadrant.

37. Graph an ordered pair on the coordinate plane.

38. Define and identify relations.

M 98 p. 5

39. Graph relations.

40. Define and identify functions.

41. Define and state the domain and range of a set.

42. Graph functions.

43. Identify linear functions.

44. Use a table of values to graph a linear function.

45. Find the y-intercept.

46. Find the slope of an equation.

47. Recognize,the slope-intercept form of an equation.

48. Identify parallel lines.

49. Find the slope of an equation by using the slopeformula.

50. Recognize the x-intercept.

51. Apply linear functions to practical situations.

52. Write linear functions from a table of values usirgthe rule method.

V. Equations and I:tequalities

Objective #5: The student will increase in knowledge ofequations and inequalities by graphingand solving sentences by completing thefollowing suggested activities to thesatisfaction of the teacher.

Activities:


53. Recognize an equation in standard form.

54. Write an equation in the satandard form ax+by+c=o.

55. Identify systems of linear equations.

M 98 p. 6

56. Solve systems of linear equations by graphing.

57. Identify equivalent systems.

58. Solve systems by substitution.

59. Solve systems by linear- combination.

60. Solve special systems of equations such as parallellines and coinciding lines.

61. Use systems of equations to solve verbal problems.

62. Write linear inequalities in slope-intercept form.

63. uraph simple linear inequalities involving, 7 ,4/... ,

;>, and 4f.. .

64. Graph compound inequalities with union and inter-sections.

65. Graph systems of linear inequalities.

66. Apply inequalities to practical situations.

VI Rational Number Operations

Objective #6: The student will increase in knowledge ofthe operations of rational numbers bycompleting the following suggested activ-ities to the satisfaction of the teacher.

Activities:


67. Identify rational expressions.

68. Simplify rational expression using a . c = ac where1; d Ed

b # 0 and d # O.

69. Multiply ational expression with:

a. monomial numerators and denominators,

b. binomial numerators and denominators, and

M 98 p. 7

c. trinomial numerators and denominators.

70. Apply the quotient rule to rational expressiona _:c=a . d

Fa o E T.

71. Divide rational expressions.

72. Find the common name for rational expressions using

a. a+ b= a+b b. a-b=a- bc c c c c c where

c # O.

73. Find the least common multiple of two or more poly-nomials.

74. Find the sums and differences of rational expres-sions with:

a. monomial denominators,

b. binomial denominators, and

c. trinomial denominators.

75. Solve equations involving rational expressions.

76. Simplify a complex fraction.

77. Apply rational expressions to practical situations.

VII. Quadratic Functions

Objective #7: The student will increase his ability toapply quadratic functions by completingthe following suggested activities to thesatisfaction of the teacher.

Activities:


78. Identify quadratic functions.

79. Find truth sets of the quadratic functions.

M 98 p. 8

80. Graph quadratic functions using a table of values.

81. Recognize the relationship between:

a. the graph of y=ax2 and the value of a,

b. the graph of y=a(x-h)2 and the value of h,

c. the oraph of y=a(x-h)2+h and the value ofh.

82. Determine the vertex and axis of symmetry of theparabola y=a(x-h)'+h.

83. Recognize the standard form of a quadratic poly-nomial.

84. Write quadratic polynomials in standard form.

85. Find the greatest and least value of a polynomialfunction.

86. Find the y-intercept of a quadratic polynomial.

87. Sketch the graph of a polynomial quadratic usingthe y-intercept, vertex, and axis of symmetry.

88. Find the zeros of quadratic polynomials from a

graph.

89. Recognize standard form of quadratic equations.

90. Write quadratic equations in standard form.

91. Find the truth set of a quadratic function by

a. factoring, and

b. quadratic formula.

92. Recognize the relationship between the Discrimenant,b -4ac, and the nature of the roots and the numberof roots.

93. Apply quadratic equations to practical situations.

M 99 p. 1

LEVEL OBJECTIVE

THE STUDENT WILL APPLY HIS KNOWLEDGE OF MATHEMATICS TO COM-PUTER PROGRAMMING BY COMPLETING THE FOLLCWING SUGGESTED AC-TIVITIES TO THE SATISFACTION OF THE TEACHER.

I. Fortran

Objective #1: The student will increase his comprehen-sion of the language and structure ofFortran by completing the following sug-gested activities to the satisfaction ofthe teacher.

Activities:


1. Define Fortran.

2. Define a computer program.

3. Define a source program.

4. Define an object program.

5. Define data.

6. Define coding.

7. Define a coding sheet.

8. Define a statement number.

9. Describe a punched card.

10. Define compiler program.

11. Define control cards.

12. Define algorithm.

13. Write a comment :ard.

14. State the Fortran character set.

15. Write basic formulas in Fortran characters.

16. Apply dilimiters to Fortran statements which in-volve addition, subtraction, division, multiplica-tion, and exponentials.

M 99 p. 2

17. Write words, numbers, names, and expressions fromthe Fortran character set.

II. Statements

Objective #2: The student will apply his knowledge ofstatements to computer programs by com-pleting the following suggested activ-ities to the satisfaction of the teacher

Activities:


18. Write an arithmetic statement which does not re-(:uire data.

19. Write an arithmetic statement which requires data.

20. Write a series of statements to solve basic formu-las of math and science.

21. Write a Read Statement which involves integer vari-ables to be used as a data set reference number.

22. Write a Read Statement which involves an integervariable and a statement number which is Formatstatement.

23. Write a Read Statement which represents a list ofone or more variable names and one or more arraynames, separated by commas.

24. Write a Format statement which will direct thecomputer to read data from punched cards.

25. Write numbers involving fixed point ariables.

26. Write a Format which allows the reading of morethan one card.

27. Write Read Statements using the Integer Format code.

28. Write Format Statements which involve Floating pointnumbers.

29. Write statements, both Read and Format, which in-volve the E and D Exponent forms.

M 99 p. 3

30. Write a WRITE Statement that involves integervariables which are used as a data set referencenumber.

31. Write a WRITE Statement that involves an integervariable and a statement number which is a FormatStatement.

32. Write a WRITE Statement which represents a list ofone or more variable names and one or more arraynames, separated by commas.

33. Write a Format Statement for the write instructionwhich will direct the computer to write variableswhich are integers or floating point.

34. Write a Format Statement for the write instructionwhich will space and label the output on the print-er of any given program.

35. Write a Format Statement which will cause the print-er to start printing on a new pane.

III. Arrays and Subscripted Variables

Objective #3: The student will apply his knowledge ofarrays and subscripted variables to com-puter statements by completing the fol-lowing suggested activities to the sat-isfaction of the teacher.

Activities:

36. Write a DIMENSION Statement which assignes namesto one or more arrays.

37. Write a subscripted variable with one, two, andthree dimension arrays.

38. Write a Format statement which causes the computerto read or write the subscripted variable.

39. Write an indexed subscripted variable which allowsthe use of only part of an array.

40. Write programs which will involve subscripted vari-ables.

M 99 p. 4

IV. Branching and Looping

Objective #4: The student will apply his knowledge ofcomputer programming by creating state-ments involving branching and looping bycompleting the following suggested activ-ities to the satisfaction of the teacher.

Activities:


41. Write a Flow Chart which will illustrate the opera-tions involved in any given program.

42. Write an IF statement which causes the computer tobranch to one of three statement numbers.

43. Write a GO TO Statement which causes the computerto branch to more than three statements.

44. Write an Unconditional GO TO Statement which allowsthe computer to return from a branch and continuethe program at a desired statement.

45. Write the DO Statement which causes the computer torepeat in a loop, operations of desired statements.

46. Write programs which involve the DO Statement tospan over an entire program.

47. Write a program which makes use of the continueStatement.

48. Write a program which involves more than one DOStatement.

V Subroutine Eubprograms

Objective #5: The student will apply his knowledge ofcomputer programming by completing thefollowing suggested activities on sub-routine subprograms to the satisfactionof the teacher.

Activities:


M 99 P. 5

49. Write the subroutine statement which allows thecomputer to solve a portion of a problem.

50. Write the Call Statement which directs the computerto branch to the subprogram.

51. Write programs which involve subroutines.

M 100 p. 1

LEVEL OBJECTIVE

THE STUDENT WILL APPLY HIS KNOWLEDGE OF THE PRINCIPLES OFMATHEMATICS TO THE OPERATION OF THE SLIDE RULE BY COMPLETINGTHE FOLLOWING LEVEL TO THE SATISFACTION OF THE TEACHER.

T. The Decimal Point

Objective #1: The student will apply his knowledge ofthe principles of decimal notation bycompleting the following suggested activ-ities to the satisfaction of the teacher.

Activities:


1. Locate the decimal point by inspection.

2. Eliminate the decimal point in products and quo-tients.

3. Write a number in scientific notation.

4. Apply the laws of exponents to multiplication anddivision.

5. Multiply numbers using scientific notation.

II. The Scales

Objective #2: The student will apply his knowledge ofthe scales on a slide rule to performingthe operations of mathematics by complet-ing the following suggested activities tothe satisfaction of the teacher.

Activities:


6. Name the physical parts of the slide rule.

7. Deterwine the difference between a uniform and non-uniform scale.

8. Describe the primary and secondary division marks onthe C and D scales.

9. Read the C and D scales for different hairline set-tings.

M 100 p. 2

10. Determine the accuracy of the different lengthslide rules.

11. Read the L scale.

12. Use the L scale to check the C and D readings.

13. Realize that a slide rule adds and subtracts lengths.

14. Multiply two numbers using the C and D scales.

15. Use either index when multiplying two numbers.

16. Divide two numbers using the C and D scales.

17. Locate the decimal point whether multiplication ordivision is performed.

18. Alternate the multiplication and division operations,regardless of which is chosen as the initial opera-tion.

19. Realize that it may be impossible at this time toalternate the division and multiplication operationson the C and D scale.

20. Identify the CI scale as being identical to the Cscale except that it reads from right to left.

21. Find the reciorocal of a number using the C and CIscales.

22. Find the reciprocal of a number using the D and DIscales.

23. Use the CI scale for division.

24. Use the CI scale for multiplication.

25. Identify the important feature of the CI scale asits treatment of multiplication as division andvice versa.

26. Use the CI scale to continue division.

27. Use the CI scale to continue multiplication.

28. Use the CI scale to continue a series of operationswhen a setting may go off the rule.

M 100 P. 3

29. Describe the CF, D, F, CIF scales and tell how theyare related to C, D, and CI scales.

30. Multiply on the folded scales.

31. Shift from the regular to the folded scales duringa combined or continued operation.

32. Shift from the folded to the regular scales duringa combined or continued operation.

33. Multiply and divide by -Til

III. Ratio and Proportion

Objective #3: The student will apply his knowledge ofthe concepts of the slide rule to solvingproblems involving ratio and proportionby completing the following suggested ac-tivities to the satisfaction of the tea-cher.

Activities:


34. Define a ratio.

35. Define a proportion.

36. Solve a proportion using the C-D (CF-DF) scales.

37. Solve an extended proportion using the C-D (CF-DF)scales.

IV. Powers and Roots

Objective #4: The student will apply his knowledge ofthe slide rule in problems of powers androots by completing the following suggest-ed activities to the satisfaction of theteacher.

Activities:

The student will be able tl:

38. Describe the A and B scales and relate their dif-ference with the C and D scales.

39. Square a number.

M 100 p. 4

40. Find the square root of a number between 1 and 100.

41. Find the square root of a whole number greater than100 or any number greater than 1.

42. Find the square root of any number less than 1.

43. Find the square root of a number using the powersof ten method.

44. Describe the K scale with its three sections.

45. Cube a number using the K scale.

46. Find the cube root of a number between 1 and 1000.

47. State the general procedure for finding cube roots.

48. Find the cube root of any number greater than 1 byusing the power of ten method.

49. Find the cube root of any number less than 1 by us-ing the power of ten method.

50. Find the 3/2 root of any number.

51. Find the 2/3 root of any number.

52. Divide a number by the square root of another num-ber using the C, D, A, and B scales.

53. Multiply a number by the square root of another num-ber using the C, 3, A, and B scales.

54. Multiply any nuober by the square of another number.

55. Divide any number by the square of another number.

56. Find the area of a circle using the A and B scales.

57. Multiply a number by the square root of another num-ber using the A and B scales.

58. Divide a number by the square root of another num-ber using the A and B scales.

59. Multiply or divide expression with square roots.

60. Multiply or divide expressions with square roots andsquares.

M 100 P. 5

61. Find the product or quotient of numbers involvingsquares, square roots or combinations of any num-ber.

V. The Trigonometric Functions

Objective #5: The student will apply his knowledge ofthe slide rule in problems involvingtrigonometric functions by completing thefollowing suggested activities to thesatisfaction of the teacher.

Activities:


62. State and identify the six basic trig functions andtheir reciprocal relations. (sin X, cos x, tan x,csc x, cot x, and sec x)

sin X = 1 /csc x

cos x = 1/sec x

tan x = 1/cot x

63. Identify the complementary relations.

(cos x = in (90-X) )

(cot x = tan (90 -x) )

(csc x = sec (90-X) )

64. Find the sin of X, where X is in degree measure.

65. Place the decimal point when using the S and STscales.

66. Find the cos of X, when X is in degree measure.

67. Place the decimal point when finding the cosine onthe S and ST scales.

68. Find the tan of X when X is a measure less than45°.

69. Find the tan of X when X is greater than 45°.

70. Place the decimal point for the tan of X for anyvalue of X.

M 100 N. 6

71. Find the cotangent of X.

72. Find the secant of X.

73. Find the cosecant of X.

74. Multiply a number and any of the six basic trigfunctions.

75. Divide a number by any of the six trig functions.

76. Describe the relation between degree measure andradian measure.

77. Convert from degree measure to radian measure andvice versa using the ST scale.

w.

M 101 p. 1

LEVEL OBJECTIVE

THE STUDENT WILL APPLY HIS KNOWLEDGE OF THE PRINCIPLES OFMATHEMATICS IN AN INTRODUCTORY COLLEGE MATHEMATICS COURSEBY COMPLETING THE FOLLOWING LEVEL TO THE SATISFACTION OFTHE TEACHER.

I. Sets, Relations, and Functions

Objective #1: The student will increase his comprehen-sion of sets, relations, and functions bycompleting the following suggested activ-ities to the satisfaction of the teacher.

Activities:


1. Define a set as any well-defined collistion of ob-jects.

2. Read the notation a EA as "a belongs to A" or "a isan element of A."

3. Distinguish between finite and infinite sets.

4. Provide examples of finite and infinite sets.

5. Describe a set by the tabulation method.

6. Describe a set by the defining-property method.

7. Recognize a mapping of a set A ''into" a set B as amatching procedure that assigns to each element "a E

A" a unique element "b 7.-. B."

8. Read the notation a -)- a' as "a maps into a'."

9. Define the set A as the domain of the mapping, theset B as the codomain of the mapping, and the setof images as the range of the mapping.

10. Recognize a mapping of a set A "onto" a set B asone in which the range is equal to the codomain.

11. Compute the cardinal number of a set.

12. Recognize two sets A and B as being equal when eachelement of A is an element of B and each element ofB is an element of A.

M 101 p. 2

13. Recognize B as a subset of A if and only if eachelement of B is an element of A.

14. List the subsets of a given set.

15. Define a set B as a proper subset of set A if andonly if B is a subset of A and at least one ele-ment of A is not an element of B.

16. Define the universe (U) as the complete set orlargest set for a particular discussion.

17. Find the solution set for a given set sekctor.

18. Define the intersection of A and B, w tten A,!B,as the set of elements common to both A and B.

19. Define the union of A and B, written A '8, as theset of elements in A or B, or in both.

20. Construct Venn Diagrams for various situations.

21. Define the null set or empty set as e set whichcontains no elements.

22. Recognize disjoint (mutually exclusive) sets as twosets having an empty intersection.

23 Define and identify denumerable sets.

24. Identify a conjunction as a compound sentence usingthe connective "and."

25. Identify a disjunction as a compound sentence usingthe connective "or."

26. Identify a negation as a sentence involving "not."

27. Construct truth tables for:

a. conjunction,

b. disjunction, and

c. negation.

28. Construct truth tables for sentences involving com-binations of conjunctions, disjunctions, and nega-tions.

M 101 p. 3

29. Identify an implication as a sentence of the form"if p then q."

30. Construct the converse, inverse, and contropositiveof a given implication.

31. Use the universal quantifier in writing open sen-tences.

32. Use the existential quantifier in writing open sen-tences.

II Real Numbers and Conditions

Objective #2: The student will apply his knowledge ofthe properties of real numbers by com-pleting the following suggested activ-ities to the satisfaction of the teacher.

Activities:


3.3. Investigate number systems as to the followingpToperties:

a. Closure,

b. Commutativity,

c. Associativity, and

d. Distributive Principle

34. Establish relationships between the various numbersystems.

35. Establish a one-to-one correspondence between thereal numbers and the points of a coordinate line.

36. Summarize the properties of the Real Number Septem.

37. Prove certain theorems concerning real numbers.

38. Perform the prime factorization of polynomials.

39. Explore the concept of order as related to a numberline.

40. Distinguish between the inclusive and exclusive "or."

M 101 p. 4

41. Solve algebraic inequalities.

42. Apply the concept of absolute value.

43. Graph sentences involving absolute value.

44. Investigate the concept of Mathematical Induction.

45. Apply Mathematical Induction to proofs.

46. Identify a field and provide examres.

47. Distinguish between upper bounds and lower bounds.

48. Explore the Completeness Property as related to theset of real numbers.

49. Identify denumerable sets.

III. Ordered Pairs, Cartesian Product ant. Relations

Objective #3: The student will increase his comprehen-sion of a Cartesian co-ordinate plane bycompleting the following suggested activ-ities to the satisfaction of the teacher.

Activities:


50. Write ordered pairs.

51. Graph ordered pairs on a coordinate plane.

52. Form a Cartesian Product.

53. Define a relation as a subset of a Cartesian Product.

54. Identify the domain of a relation as the set of allfirst components of the ordered pairs in the orderedpairs in tne relatior-

55. Identify the range of a relation as the set of allsecond components of the ordered pairs in the rela-tion.

56. Identify an equivalence relation as a relation whichexhibits the following properties:

a. Reflexive.

b. Symmetric, and

c. Transitive.

M 101 D. 5

57. Provide examples of relations which may be classi-fied as equivalence relations.

58. Identify pairs of complementary relations.

59. Construct the inverse of a relation.

IV. Relations and Functions

Objective #4: The student will apply his knowledge ofthe principles of relations and functionsto graphing by completing the followingsuggested activities to the satisfactionof the teacher.

Activities:


60. Graph examples of relations on a coordinate plane.

61. Graph inequalities in 4,o variables.

62. Graph relations involving compound conditions.

63. Define a function as a relation in which no twoordered pairs have the same first compo'ient.

64. Use functional notation to evaluate a function ata particular value of X.

65. Identify a neighborhood or a deleted neighborhocdof X.

66. Apply the operations of union and intersection torelations and functions.

67. Perform the operations of addition, subtraction,multiplication, and division with functions.

68. Construct the inverse of a function.

69. Perform the composition of two functions.

V. Numerals and Numeration

Objective #5: The student will apply hs knowledge ofthe properties and operations of the realnumuer system by completing the followingsuggested activities to the satisfactionof the teacher.

M 101 a. 6

Activities:


70. Distinguish between numeral and number.

71. Represent numbers in the Roman system of numeration.

72. Represent numbers in the Egyptian system of numera-tion.

73. Represent numbers in the decimal system of numera-tion.

74. Write expanded numerals in the decimal system ofnumeration.

75. Represent numbers in non-decimal systems of numera-tion systems.

76. Perform operations of addition, subtraction, multi-plication, and division in non-decimal numerationsystems.

77. Represent fractions in non-decimal numeration sys-tems.

78. Define a natural number.

79. Perform operations of addition and multiplicationinvolving natural numbers.

80. Perform operations of subtraction and division ofnatural numbers when a result exists.

81. Determine the property of closure when it existsfor an operation on the set of naturals.

82. Demonstrate the commutative laws for addition andmultiplication for naturals.

83. Demonstrate the distributive law for multiplicationover addition for naturals.

84. Demonstrate the associative laws for addition andmultiplication for naturals.

85. Demonstrate that the addition algorithm depends onthe commutative andassociative laws for addition.

M 101 p. 7

86. Demonstrate that the multiplication algoritilm de-pends on the distributive law of multiplicationover addition.

E17. Identify prime numbers.

88. Factor any composite number into a product ofprimes.

89. Find the least common multiple for two or more nat-ural numbers.

90. Find the greatest common factor for two or more nat-ural numbers.

91. Identify even natural numbers.

92. Identify odd natural numbers.

93. Find figurate numbers including:

a. oblong,

b. rectangular,

c. triangular,

d. square, and

e. cube numbers.

94. Determine divisibility by the numbers 3, 4, 5, 6, 8,9, and 10.

95. Check the operation of addition by casting out nines.

VI. Mathematical Systems

Objective #6: The student will apply his knowledge ofthe principles of the real number systemto any mathematical system by completingthe following suggested activities to thesatisfaction of the teacher.

Activities:


96. Examine and construct mathematical systems otherthan natural numbers, including

a. Clock authmetic,

M 101 p. 8

b. modular arithmetics, and

,:. mathematical systems without numbers.

97. Determine operational identity for a set.

98. Determine operational inverses for a set.

99. Determine commutative porperty for a set and opera-tion.

100. Determine an associative property for a set andoperation.

101. Determine that a set and an operation defined on aset form a group or an abelian group.

102. Determine a subgroup.

103. Determine a distributive law for a set and two oper-ations defined on the set.

104. Determine that a set and two operations defined onthe set form a field.

105. Determine that a set and two operations defined onthe set form a ring.

VII Mathematical Reasoning and Creativity

Objective #7: The student will be able to analyze mathe-matical data fn.- proofs by completing thefollowing suggested activities to the sat-:sfaction of the teacher.

Activities:


106. Arrive at a conclusion inductively through examiningcertain samples.

107. Arrive at a conclusion through deductive reasoning.

108. Perform proofs in a mathematical system.

109. Use diagrams to illustrate correct reasoning.

110. Establish the vali1ity of a statement.

M 101 P. 9

111. Distinguish between a statement and a sentence.

112. Use variables, constants, and symbols in sentences.

113. Identify a replacement set.

114. Identify a solution set as a subset of some replace-ment set.

115. Express sets using set notation.

116. Identify set notation.

117. Represent set relations symbolically, including:

a. intersection,

b. subset (proper or improper),

c. union, and

d. complement.

118. Rep esent the following set relations using dia-grams:

a. subset,

b. intersection,

c. union, and

d. complement.

119. Determine universal sets.

120. Determine equality for sets.

121. Determine the cardinal numbers of sets.

122. Determine a one-to-one correspondence.

123. Define a number in terms of sets.

124. Determine truth value for compound statements,spicifically

a. the conjunction,

b. the disjunction,

c. the conditional, and

d. the biconditional.

M 101 p. 10

125. Form the converse, inverse and contrapositiv,.: of agiven implication.

126. Form the negation of a given statement.

127. Determine truth value for compound sentences usingmore than one connective.

128. Prove a compound sentence is a tauto;ogy by a truthtable.

129. Form negations of compound sentences.

130. Write a definition.

131. Structure a statement using a quantifier (universalor existential) and a sentence.

132. Structure a statement u_ing more than one quanti-fier.

133. Form the negation of quantified statements.

134. Use sentences to describe sets.

135. List the characteristics of Boolean Algebras.

136. Use rules of inference to arrive at desired conclu-sions including:

a. substitution,

h. niodus ponens,

c. tautologies, and

d. rule c: unditional proof.

137. Test the validity of an argument.

138. Prove certain theorems about groups.

139. Prove certain theorems about fields.

140. Write an informal proof.

141. Use mathematical induction to form a conclusion.

M 102 p. 1

LEVEL OBJECTIVE

THE STUDENT WILL APPLY HIS KNOWLEDGE OF ALGEBRAIC 'TINCIPLESBY COMPLETING THE Fo_LOWING LEVEL TO THE SATISFAr LON OF THETEACHER.

I. Algebraic Expressions

Objective #1: The student will apply his knowledge ofalgebraic expressions by completing thefollowing suggested activities to thesatisfaction of the teacher.

7 Activities:


1. Identify a given numeral as natural, whole, ration-al, or irrational.

2. Simplify a given numerical phrase using propertiesof numbers.

3. Describe the sturcture of the real number system.

4. Express a given periodic decimal as a rational num-ber.

Find the solution set of a linear equation.

6. Recognize equivalent equations.

7. Arrive at an equivalent equation from a given .ine-ar equation to find the solution set by transforma-tion.

8. Find the distance between two points from the co-ordinates of the points.

9. Derive the distance formula.

10. Compute expressions involving positive integral ex-ponents.

11. Compute expressions involving negative integral ex-ponents.

12. compute expression involving rational exponents.

13. Relate fractional exponents and radicals.

___

M 102 p. 2

14. Express a radical in simplest form.

15. Perform the operations of addition, multiplication,division, and subtraction with radicals.

16. Simplify expressions involving rational exponents.

17. Factor algebraic expressions such as:

a. common monomial factoring,

b. common binomial factoring,

c. difference between 2 perfect squares,

d. perfect trinomial square,

e. general trinomial, and

f. an + bn.

18. Perform operations involving fractions.

19. Simplify algebraic fractions.

20. Add and subtract algebraic fractions.


22. Solve fractional equations in one variable.

II. Quadratic Equations

Objective #2: The student will apply his knowledge ofalgebraic principles to quadratic equa-tions by completing the following sug-gested activities to the satisfactionof the teacher.

Activities:


23. Solve a quadratic equation using the factoring meth-od.

24. Solve a quadratic equation by completing the square.

25. Sol', i quadratic equation using the quadraticforLula

M 102 D. 3

26. Graph a quadratic equation.

27. Determine the nature of the roots of a quadraticequation by observing the discriminant.

28. Relate the sum and product of the roots to a qua-dratic equation in standard form.

29. Find the quadratic equation given the roots of aquadratic equation.

III. Logarithms

Objective #3: The student will apply his knowledge oflogarithms to the appropriate mathemat-ical operations by completing the fol-lowing suggested activities to the sat-isfaction of the teacher.

Activities:


30. Express a logarithmic expression as an exponentialexpression.

31. Apply exponential laws to properties of logarithms.

32. Apply properties of logarithms to any base.

33. Solve equations involving logarithms.

34. Explain the development of the common logarithms.

35. Use a table of hantissas to find the common logar-ithm of a real number.

36. Use a table of mantissas to Find antilogarithm.

37. Use the nettas of linear interpolation to approxi-mate manii:sas and antilogarithms not found in atable.

38. Use common logarithms to perform the operations ofmultiplication, division, raising to powers and ex-tracting roots.

39. Solve exponential equations using common logaritf- ,.

M 102 p. 4

IV. Mathematical Sequences

Objective #4: The student will apply his knowledge ofmathematical sequences to problem solv-ing by completing the following suggest-ed activities to the satisfaction of theteacher.

Activities:


40. Identify an arithmetic seq nce of numbers.

41. Find the nth term of an arithmetic series.

42. Find the sum of the first n terms in an arithmeticseries.

43. Insert arithmetic means between two extremes.

44. Identify a geometric sequence of numbers.

45. Find the nth term of a geometric series.

46. Find the sum of the first n terms of a geometricseries.

47. Insert geometric means between two extremes.

V. Statistics and Probability

Objective #5: The student will apply his knowledge ofstatistics and probability to problemsolving situations by completing thefollowing suggested activities to thesatisfaction of the teacher.

Activities:


48. Identify permutations.

49. Solve problems involving permutations.

50. Identify combinations.

51. Solve problems involving combinations.

52. Find the probability an event will occur.

M 102 p. 5

53. Find the probability an event will not occur.

54. Find the probability two or more events will occur.

55. Identify mutually exclusive events.

VI. Theory of Equations

Objective #6: The student will apply the theory of equa-tions to solving problems by completingthe following suggested activities to thesatisfaction of the teacher.

Activities:


56. Identify systems of equations in two or three vari-ables.

57. Classify systems as linear-linear, linear-quadraticor quadratic-quadratic systems.

58. Find solution sets for systems of equations usingthe multiplication-addition method or the substitu-tion method

59. Solve systems of equations graphically.

60. Solve systems of inequalities algebraically.

61. Solve systems of inequalities graphically.

62. Solve systems of equations using determinants.

63. Use synthetic division to divide a polynomial by abinomial.

64. Use synthetic division to factor a polynomial.

65. Use the Factor Theorem to factor a polynomial.

66. Use the remainder Theorem to divide a polynomial bya binomial.

67. Graph equations of a higher than second degree.

VII. Curve Sketching

Objective #7: The student will apply his knowledge ofgeometric interpretation to an algebraic

p. 6

situation by completing the followingsuggested activities to the satisfac-tion of the teacher.

Activities:


68. Graph a linear equation by the slope-interceptmethod.

69. Graph a linear equation by using the x and y-inter-cept method.

70. Find the y-intercept and x-intercept of a graph.

71. Graph a parabola.

72. Determine the axis of symmetry of a parabola.

73. Determine the translation of axis of a parabola.

74. Find the maximum or minimum points of a parabola.

75. Solve verbal problems involving maximum and minimumpoints of a parabola.

76. Graph a circle.

77. Determine the Translation of Axis for a circle.

78. Graph an ellipse.

79. Determine the translation of axis of an ellipse.

80. Graph a Hyperbola.

81. Determine the translation of axis of a hyperbola.

82. Graph an exponential function.

83. Graph a linear inequality.

84. Graph the intersection and union of linear inequal-ities.

85. Graph inequalities of the four conics.

86. Determine the discontinuity of a function.

M 103

LEVEL OBJECTIVE

D. 1

THE STUDENT WILL APPLY HIS KNOWLEDGE OF THE PRINCIPLES OFGEOMETRY BY COMPLETING THE FOLLOWING LEVEL TO THE SATIS-FACTION OF THE TEACHER.

I. Triangles and Quadrilaterals

Objective #1: The student will apply his knowledge ofthe properties of triangles and quadri-laterals to theorems and proofs by com-pleting the following suggested activ-ities to the satisfaction of the teacher.

Activities:


1. Define basic terms which are used in the study ofgeometry, such as: midpoint, angle bisector, per-pendicular, right angle, acute, adjacent angles,vertical angles, complementary angles, and supple-mentary angles.

2. Classify triangles by aides.

3. Classify triangles by angles.

4. Define and apply lines and segments associated withtriangles, such as altitude, median, and perpendic-ular bisector.

5. Solve numerical problems involving angles.

6. Define exiom and postulate and know the importantaxioms and postulate related to segments and angles.

7. Know the definition of Theorem and know those theo-rems which apply to vertical supplementary, andcomplementary angles.

8. Select the hypothesis (given) and conclusion (toprove) from a geometric statement.

9. Write the converse of a theorem.

10. Apply those theorems concerned with isosceles tri-angles.

11. Apply those theorems concerned with parallel lines.

M 103 p. 2

12. Apply those theirems concerned with proving trian-gles congruent.

13. Learn the definitions and properties of the varioustypes of quadrilaterals.

14. Apply the concept of distance.

15. Know and prove when a quadrilateral in a parallela-gram, rectangle, square, or rhombus.

16. Identify the properties of all the quadrilaterals.

17. Know those theorems concerning distance with per-pendicular lines and bisectors.

18. Know those theorems which prove lines are parallel.

19. Know and apply those theorems and corollaries whichconcern the measures of the angles of a triangle orquadrilateral.

20. Apply those theorems and definitions which concernthe exterior angles of a triangle.

21. Find the measure of an interior or exterior angleof a regular polygram.

II. Circles

Objective #2: The student will apply his knowledge oftheorems concerning circles by completingthe following suggested activities to thesatisfaction of the teacher.

Activities:


22. Identify and name the various parts of a circle andsegments or lines associated with one or more cir-cles.

23. Understand the relationships between central anglesand arcs.

24. Explain the difference between the length of an arcand the measure of an arc.

25. Distinguish between inscribed and circumscribedpolygrams.

M 103 p. 3

26. Apply the theorems and corollaries concerning in-scribed angles of a circle.

27. Use the term secant ray and tangent ray in rela-tions to describing the positions of angles tocircles.

28. Apply those theorems concerning chords, secants,and tangents.

III. Ratio and Proportions: Similar Triangles

Objective #3: The student will apply his knowledge ofratios and proportions to similar tri-angles by completing the following sug-gested activities to the satisfactionof the teacher.

Activities:


29. Define ratio and proportion.

30. Identify the properties of a proportion.

31. Know how to prove triangles similar.

32. Apply the relationships between segments of chords,secant segments, andtangent segments.

33. Apply the relationships between the altitude to thehypotenuse and the segments of the hypotenuse of aright triangle.

34. Apply the relationships among the legs and thehypotenuse of a right triangle.

35. Apply those theorems concerning lines parallel toone side of a triangle.

36. Apply those theorems concerned with bisecting anangle of a triangle.

37. Apply those ratios which concern corresponding sidesof similar triangles.

38. Apply those theorems which concern ratios of chordsand their segment, tangent segments and secant seg-ments.

M 103 p. 4

39. Understand and apply the Pythogorean Theorem totriangles, circles, and polygons.

40. Solve quadratic equations which may develop fromthese special ratios and theorems.

41. Find the lengths of the sides of a 30-60 right tri-angle.

IV. Areas of Polygons

Objective #4: The student will apply his knowledge ofthe concept of area to various polygonsby completing the following suggestedactivities to the satisfaction of theteacher.

Activities:


42. Apply those theorems concerning areas of triangles,parallelograms, rectangles, and trapezoids.

43. Apply the Pythogorean Theorem to finding the areasof triangles and quadrilaterals.

44. Apply the special ratios of 30-60 right trianglesto find the areas of triangles and similar poly-gons.

45. Apply those theorems concerning areas of similartriangles and similar polyms.

46. Find the area of a regular polygon.

47. Find the sum of interior and exterior angles ofpolygons.

48. Apply those theorems concerned with sides, perime-ters, and areas of similar polygons.

49. Find the circumference and area of a circle.

50. Apply those ratios which concern radis, diameter,circumferences, and areas of circles.

51. Find the length of arc, area of a sector, and thearea of a segment of a circle.

M 103 p. 5

V. Inequalities and Locus

Objective #5: The student will apply his knowledge ofthe principles of inequalities and locusto problem solving by completing the fol-lowing suggested activities to the sat-isfaction of the teacher.

Activities:


52. Apply those axioms and theorems concerning inequali-ties and inequalities of triangles.

53. Solve those problems in solving proofs of inequali-ties and computational problems.

,54. Know the. meaning Locus.

55. Determine the Locus of points concerning circles,parallel lines, perpendicular and perpendicular bi-sectors, angle bisectors, and triangles.

56. Know the meaning of intersection of two loci.

57. Determine the intersection of two loci.

58. Use the preparing for the college boards text totest comprehension of subject matter.

M 104 p. 2

6. Construct the graph of the sine and cosine curve byusing the unit circle and justify why both curvesmust be continuous and periodic.

7. Describe thebehavior of sine x and cosine x overgiven intervals.

8. Find the sin x, given the cos x where x is not aspecial real.

9. Derive the following reduction formulas:

sin /7: ± cos (7i* z

sin t/,;Irr 4) cos

sin (-3e, cos

by a. using the symmetries of the unit circle,

b. graphing using equations of translationsand solve related problems.

10. Derive sin (r -x) = cos x and cos (7- -x) = sin x byusing: In the same circle, equal chords have equalareas.

11. Derive the reduction formulas

sin 1(2n + 1) +x]where n nonegative in-

cos [(2n + 1) +x] tegers

by a. using symmetries to tf -x on unit circle,

b. graphing using equations of translations.

and solve related problems.

12. Derive the following formulas:

cos ( xl -x2)

cos ( xl + x2)

sin ( xl t x2)

sin 2x

cos 2x

:sin 1/2 x 1

!cos 1/2 x 1

M 104 P. 3

and solve related problems.

13. Graph on real plane

y = a sin (bx + e) + d = a sin bx'

y = a cos (bx + e) + d = a cos bx'

by using translation of axis.

14. Write an equation of a sinusoidal curve given am-plitude, period and phase shift.

15. Graph the sum of two sinusoidal curves by ordinateaddition.

II The Tangent, Cotangent, Secant, and Cosecant Functions

Objective #2: The student will apply his knowledge ofall six trigonometric functions to deri-vations and problem solving by completingthe following suggested activities to thesatisfaction of the teacher.

Activities:


16. Define the tangent, cotangents secant, and cosecantfunctions specifying domain and range.

17. Evaluate the circular functions of the special realsmentioned in #le and #3.

18. Determine all circular functions from one given cir-cular function.

19. Graph over the real plane x--.bf(x) where f(x) etan, cot, sec, cscj

20. Derive the following reduction formulas:

f [( 2n + 1 )it + x] nE I2

by:

where f = (tan, cot,f En 71- = x] ne* even I sec, csc3

2

a. using definition of functions and symme-tries of unit circle.

b. graphing using equations of translationsand solve related problems.

M 104 p. 4

21. Derive the following formula and solve relatedproblems:

tan 2(x1 + x2)

tan 2x

tan 1 x

22. Define the converse of each of 4:he circular func-tions specifying the domain and range and grapheach of the relations.

23. Explain why: a. it is necessary to: restrict thedomain of circular functions for the existence ofinverses.

b. the particular domains are sel-ected.

24. Define and graph the inverses of the circular func-tions.

25. Derive the following formulas and apply them.

sin (Sin' u) = u-1ros [Tan u]

cos (Cos-I u) = u tan [Sin -' u]/..

sin (Cos-I u) tan [Cos u] = 1.1"

cos [Sin--1 u] Cot -I u--Tan-1( )

where 0

sin [Tan-' u]

...

III. Angles and Arcs

Sec-lu---Cos-1("1),where w # o

Objective #3: The student will apply his knowledge oftrigonometric functions to situations in-volving angles and arcs by completing thefollowing suggested activities to thesatisfaction of the teacher.

Activities:


M 104 p. 5

26. Solve for a particular or a general solution of anopen sentence that may require a substitution ofany previously proven identity.

27. Derive and apply the sum and product formulas fromthe sum and difference formulas of the sine and co-sine.

28. Distinguish between the geometric and trigonometricdefinition of an angle.

29. On the coordinate plane, recognize angles in stan-dard position, coterminal angles, positive and neg-ative angles.

30. Distinguish between the circular functions and thetrigonometric functions.

31. Illustrate the correspondence from the coordinatesof points x on the real number line to points P onthe unit circle to angles 4:AOP in standardposition.

OID A32. Illustrate why it is natural to define sin AOP to

be the ordinate of P and cost AOP to be the abscissaof P.

33. Draw a diagram of angles in standard position in theunit circle and find the sine and cosine of the an-gles which correspond to the special reals.

34. Determine the sine and cosine of the angle in stan-dard position given the coordinates of a point onthe terminal side using similar triangles and theunit circle.

35. Prove that, if (a,b) are the coordinates of a pointon the terminal side of an angle in standard posi-tion then

sin4A0P = b

a2 + b2 # 0 a2 + b2costA0P =

a + b2

36. Determine the formulas for the remaining functionsby using the results in #35.

37. Demonstrate the isomorphism between the CircularFunctions and the Trigonometric Functions by show-ing that:

M 104

a. Definitions Correspond

Circular Trigonometric

Real number on unit Angle in stan-circle dard position

Primary Real Number Primary Angle

.a.0 set of reals Domain set ofangles

b. Operations are Pre-served

p. 6

38. Demonstrate how angle measure is related to circu-lar arc measure by using the teal number line andthe unit circle to find the unit are on circle C.

Unit are on C = 2 where h = units con-h tained in the circle.

39. Define the following units of angle measurc.: de-gree, radian, centangle.

40. Show that the relationship between w, the degreemeasure of an angle and y the radian measure ofthat angle is expressed by:

Tr- = y180 w

41. Use the proportion from #40 to convert from degreemeasure to radian measure and vice versa.

42. State the length of an arc intercepted by a givencentral angle and state the measure of the centralangle intercepted by an arc with a given lengtr.

43. Use a table of values of trigonometric functions tofind:

a. the value of trigonometric functions

b. angles from a given numerical value of atrigonometric function of some angle.

44. Solve word problems that require construction ofright triangles and computations with trigonometricfunctions.

M 104 p. 7

45. Use the Cstance formula to derive the Law of Co-sines and apply the Law of Cosines in solving prob-lems.

46. Derive and find the area of a triangle given twosides and the included angle of a triangle.

47. Derive the Law of Sines and use this law to solverelated problems.

48. Express complex numbers in polar or trigonometricform.

49. Perform the operations of addition, subtraction,multiplication, and division of complex numbers inpolar form.

50. Solve equations using De Moivre's Theorem.

M 105 p. 1

LEVEL OBJECTIVE

THE STUDENT WILL APPLY HIS KNOWLEDGE OF ALGEBRAIC PRINCIPLESBY COMPLETING THE FOLLOWING LEVEL TO THE SATISFACTION OF THETEACHER.

I. Mathematical Statements

Objective #1: The student will apply his knowledge ofthe principles of logic by completing thefollowing suggested activities to thesatisfaction of the teacher.

Activities:


1. Distinguish between a sentence and a statement.

2. Define variable, replacement set, and open sentence.

3. Translate simple mathematical sentences using theuniversal and existential quantifiers.

4. Assign truth values to open sentences.

5. Write generalizations symbolically - using quanti-fiers, open sentences, and at most four variables -when given a set of statements.

6. Assign truth values to each of the following state-ments:

()JP p q

o y q p q

P A q

7. Assign truth values to composite statements usingthe basic truth tables given in #6 and at most fourvariables.

8. Negate simple, composite, and quantified statementssuch as:

(P) n. ( )

P V q ) ti [4V1 3 ;,p)]

r. ( p q ) (P)]

M 105 p. 2

9. Identify logically valid statements, logicallyequivalent statements and contradictory statementsby constructing truth tables.

10. Identify valid statements by showing that the con-clusion logically follows from the premises andrecognize certain valid and invalid forms.

11. Write the conditional in the following forms:

If p then q

p only if q

q is a necessary conditionfor p

p is a sufficient conditionfor q

12. Write the variants of the conditional: the converse,inverse, and contrapositive.

1 . P v ei I> -4") e > 9 /Null ict c v c/ 'I " 12 ' civn.td. ig______ _f____- ......,....ct ..............._

... ______ z:te,___:___.

P 1 .1.. r P p 1,,9

13. Write the biconditional in the following forms:

If p then q and if q then p

p if and only if q

p is a necessary and sufficient condition for q

14. Recognize the difference between proof by contra -di cti on(p -0.q)e-aNpAl and the indirect proof.k

15. Illustrate by example how advertisers and propagan-dists make use of logical equivalence and invalidinference patterns.

16. Recognize examples of reflexive, symmetric, andtransitive properties.

17. Define equivalence relation and recognize examplesequivalence relations.

18. Use the elements of logic in mathematical proofs:

(x=y)---0(x+z = y+z)

(x=y).---1-(x-z = y-z)

(x=y) = yz) -

[(z$0) A (x=.0]> (x =(z z)

M 105 p. 3

Ex=y)/;(z=w)]--*(x+z = y+w)

Ex=y)A(z=w)]--1,-(xz = yw)

II. Real Number Properties

Objective #2: The student will increase his comprehen-sion of the real number system by com-pleting the following suggested activ-ities to the satisfaction of the teacher.

Activities:


19. Define binary operation and recognize rules of com-bination defined over subsets of the reals which arebinary operations.

20. Recognize examples of the following properties; andgiven an example over a subset of the reals, recog-nize whether the following properties hold for addi-tion and multiplication:

a. Closure

b. Associative

c. Existence of Unique Identity Elements

d. Existence of Unique Inverse for Each Ele-ment

21. Demonstrate the properties of closure, associativ-ity, identity, and inverse for modular arithmeticover multiplication and addition.

22. Demonstrate and describe the relation of "primeness"of module to the existence of a multiplicative andadditive inverse.

23. Define a group and recognize whether a set definedover an operation is a group.

24. Recognize examples of commutative property and wheth-er this property holds for the operations of multi-plication and addition.

25. Define Abelian (Commutative) Group and recognizewhether a set defined over an operation is an Abel-ian Group.

M 105 p. 4

III. Proofs

Objective #3: The student will increase his ability toanalyze given information to develop math-ematical proofs by completing the follow-ing suggested activities to the satisfac-tion of the teacher.

Activities:


26. Write the proofs of the following statements:

X) `j, z G 5A Z=w--* X-*Z zE Gi x,rz= ti z x

27. Recognize examples of the distributive property ofmultiplication over addition and whether it holdsfor subsets of the reals.

28. Define field and recognize whether a set definedover the operation of addition and multiplicationis a field.

29. Write the axiom, theorem, or definition justifyingsteps in proofs of properties of real numbers.

30. Write formal proofs of the following statements:

(a = b) --.(a + e = b + c) (a = 0-,)(as =be)

[(a = b) A (c = + e = b + d)[(a = b)A (e =

(b + e) = ( -c) = b d)] = bd)

(a + b = 0)--10'(b = -a) (be)1 = b ife

-(a + b) = (-a) + (-b) e # 0

a ° 0 = 0s a = 0

(-a) (b) = -ab

(x # b = a) -.'(x = a+(-b) )

1 = 1 . 1

-a-- a E

a(-1) = (-1)a = -a

(-a) (-b) =-ab

P. 5

xb = a-->x =a . 1 (b#0)

b

31. Write formal proofs of theorems showing the rela-tion between addition and subtraction, and the rela-tion between multiplication and division.

32. Apply the properties of a field, theorems on addi-tion, multiplication, subtraction, and division toevaluating expressions.

33. Write formal proofs of the following statements:

[(0Y)A (Y z)](x)z)

(x)Y)-4(x + z) > (Y + z)]

r(x>y)it (z6 e)]rxz'yz

[(x7y) A (z E R-)].-+XZcyZ

[(k<y)A (ycz)]-4.(x z)

(z)]x4y)[(x + z)4(y +

Nkey)A (z( R+)]-->xz.eyz

[(x4y)A (zE R-)]-..xz>yz

34. Establish the density of the real numbers by find-ing a real number between any two real numbers.

35. Define completeness and a Complete Ordered Field.

M 106 p. 1

LEVEL OBJECTIVE

THE STUDENT WILL INCREASE IN HIS ABILITY TO ANALYZE THE PROP-ERTIES OF THE REAL NUMBER SYSTEM BY COMPLETING THE FOLLOWINGLEVEL TO THE SATISFACTION OF THE TEACHER.

I. Rational Expressions

Objective #1: The student will increase his ability toanalyze the rational number concepts todevelop proofs by completing the followingsuggested activities to the satisfactionof the teacher.

Activities:

The stu,!.1nt will be able to:

1. Define and identify polynomials.

2. Perform basic operations of addition, subtraction,multiplication, and division of polynomials by us-ing the laws of the field.

3. Use the Unique Factorization Theorem for Polynomialsto completely factor polynomials which may be in thefollowing forms:

a. zlix + zy = z(x + y) e. acx2 + (ad + bc)x + bd = (ax +b)

b. x2 + 2xy + y2 = (x + y)2 (cx+d)

c. x2 - 2xy + y2 = (x - y)2 f. ax + ay + bx + by =(a + h)(x + y)

g. x 3 4' Y3 = (x + y)(x2 -xy + y2)

h. x3 - y3 = (x - y)(x2 + xy + y2)

d. x2 - y2 = (x + y)(x - y)

4. Use the Factor Theorem and the Remainder Theorem tofactor polynomials not in any of the above forms.

5. Define and identify rational expressions.

6. Perform basic operations of additioi, subtraction,multiplication, and division with rational expon-ents.

M 106 p. 2

7. Define zero exponents and negative exponents asfollows:

a # 0, a0 = 1.

If a # 0, = 1 1 = aN where n E Reals+an and a-n

8. Define simplest form of rational expressions tomean no zero and no negative exponents.

9. Write formal ;proofs of: Aix,/ e ReGis 4 r"( Vin, 2: IT&

a.Z = Xrl

C r 1

d. =7' j X 00

e. 5fx.7

V 0P_

10. Define fractional exponents as aq =

II. Irrational Expressions

Objective #2: The student will increase his ability toanalyze the irrational number concepts todevelop proofs by completing the follow-ing suggested activities to the satisfac-tion of the teacher.

Activities:


11. Define simplest form of Irrational expression tomean no fractional exponents and no fractions asradicands and no radicals in the denominator.

12. Write formal proofs of the following: Yph,,,, E- Imc5c,.1: ./,-ch., ),J...,... (, )199 ...1- i, 7-I, ) 1""P V E" A) I 4jr, 1 en ...s

a . fx r% 1" X 31-0 k t 1 1..) ( 4 PJ

t I ir 1 r--- - r 1 ,.. _.... 1)

b . ("X y )3; -z 'X 7-:% i-j." -y A- t) --- -v ,,k- -1/77-

... 12 "; -- /1C . 1.2- ' 7 " -h i ±

-X -711 rl li X 75 =ps. i .

-- 70"2--,:,n t X. 1,77, -yx -:- V A - .1, ,,-, , I

M 106 P. 3

13. Perform basic operations of addition, subtraction,multiplication, and division with Irrational ex-pressions.

14. Use the properties of the Field to solve first de-gree equations in one variable.

15. Use properties of Ordered Field to solve inequali-ties in one variable.

16. Solve equations and inequalities in one variableinvolving absolute value.

17. Write formal proofs of the following:

a i X 6 Req1.5 191?.. X and 121 -?: - Xb . Aix, tj E 1?e,,i; I'Xi + 1 1 -?: I 4 i

c. 41%,5 E Re,IS i X') =-; ixi ` hitd.-Vx,e,e K 121 < c" (-4 0- < c ct-e Vz, a Lc / 1') > a (-) x z - 4. or, > a_

M 107 D. 1

LEVEL OBJECTIVE

THE STUDENT WILL APPLY HIS KNOWLEDGE OF THE PRINCIPLES OFTHE REAL NUMBER SYSTEM BY COMPLETING THE FOLLOWING LEVEL TOTHE SATISFACTION OF THE TEACHER.

I. Functions

Objective #1: The student will apply his knowledge ofthe principles of functions and equationsto finding solution sets by completingthe following suggested activities to thesatisfaction of the teacher.

Activities:


1. Define and write the Cartesian Product of two sets.

2. Define and identify relation, R, from one set toanother set.

3. Define and identify the Domain and Range of a rela7tion.

4. Define and identify function as Function, F, fromset A to set B is a relation from A to B such that

1. A is the domain of F

2. If (a,b) F and (z,c) - F, then b = c.

5. Differentiate between into, onto, and one-to-onemappings.

6. Use function notation: f = [(x,y)/y = f(x)] wheref(x) is the image of x.

7. Differentiate between an even and an odd function.

8. Relate even and odd functions to symmetry.

9. Define and identify periodic functions as f(x) .f(x + h).

10. Perform the addition, subtraction, multiplication,and division of functions.

11. Define and form the composition of functions.

M 107 p. 2

12. Form the converse of a function and indicate whetherthe converse is a function called the inverse.

13. Indicate whether a function is increasing, decreas-ing, constant, or neither over a particular inter-val.

14. Define linear function.

15. Define and write the inverse of a linear function.

16. Define and find the zero of a linear function.

17. Graph linear functions using slope, intercepts,and other points.

18. Graph and write the equation of linear functionsparallel to the given function.

19. Graph and write the equation of a linear functionperpendicular to the given function.

20. Define quadratic function and differentiate betweenlinear and quadratic functions.

21. Find the solution set of the quadratic function,f(x) = 0, by factoring, completing the square, andthe quadratic formula.

22. Find the discriminant to identify the type of solu-tion of a quadratic equation.

23. Relate the sum and product of the roots to the co-efficients of a quadratic function.

24. Write the equation of a quadratic function usingthe sum and product of the roots.

25. Solve quadratic inequalities.

26. Solve irrational equations, checking for extrane-ous roots.

II. Conic Sections

Objective #2: The student will increase his comprehen-sion of conic sections by completing thefollowing suggested activities to thesatisfaction of the teacher.

M 107 p. 3

Activities:


27. Define circle.

28. Define parabola.

29. Define ellipse.

30. Define hyperbola.

31. Derive the equation for the above conic sectionsat the origin and translated to the point (h,k).

32. Recognize the properties as they relate to eachconic:

a. vertex

b. axis of symmetry

c. focus

d. latus rectum

e. eccentricity

f. directrix

III. Logarithms

Objective #3: The student will apply his knowledge oflogarithms to mathematical operations bycompleting the following suggested activ-ities to the satisfaction of the teacher.

Activities:


33. Define the exponential function.

34. Graph the exponential function.

35. Define the logarithmic function as the inverse ofthe exponential function.

36. Differentiate between common logarithms and logar-ithms with other bases,

M 107 p. 4

37. Use the properties of logarithms to perform mul-tiplication, division, powers, and roots.

38. Use linear interpolation to gain a greater degreeof accuracy.

IV. Complex Numbers

Objective #4: The student will apply his knowledge ofthe principles of a numeration system toinclude complex numbers by completing thefollowing suggested activities to the sat-isfaction of the teacher.

Activities:


39. Define complex numbers as a + bi and (a,b).

40. Recognize and prove the Field Properties for theset of complex numbers:

a. Identify of addition and multiplication

b. Inverses for addition: and multiplication

c. Closure cf addition and multiplication

d. Associativity of addition and multiplica-tion

e. Commutativity for addition and multiplica-tion

f. Distributive of multiplication over addi-tion

41. Perform the basic operations of addition, multipli-cation, subtraction, and division with complex num-bers.

42. Simplify comple> numbers using the Field Propertiesand the conjugate.

43. Graph a complex number.

44. Perform the operations of addition and subtractiongraphically.

M 107 p. 5

V. Polynomials and Sequences

Objective #5: The student will apply his knowledge o4mathematical reasoning to polynomials acidnumber sequences by completing the fol-lowing suggested activities to the sat-isfaction of the teacher.

Activities:


45. Define polynomial function.

46. Define rational function.

47. Use synthetic division to shorten division problems.

48. Utilize the Factor Theorem, Remainder Theorem, andSynthetic Division to factorhigher degree polynom-ials.

49. Utilize descartes Rule of Signs to solve higher de-gree polynomial functions.

50. Find the Complex zero of real polynomials.

51. Graph polynomial functions.

52. Use Mathematical Induction as a method of proof.

53. Define and recognize Arithmetic sequences.

54. Define first term, last term, nth term, and commondifference.

55. Derive and the formula for finding the first term,nth term, number of terms, or the common differencefor an Arithmetic Sequence.

56. Define and find arithmetic means between any twoterms of a sequence.

Li. Define and derive a formula for tie sum of an Arith-metic Series.

58. Define and recognize Geoemtric Sequences.

59. Define first term, last term, nth term, and commonfactor.

M 107 p. 6

60. Derive and use the formula for finding the firstterm, nth term, number of terms, or common factor.

61. Define and find Geometric means between any twoterms of a sequence.

62. Define and derive the formulas for the sum of afinite and infinite Geometric Series.

M 108 p. 1

LEVEL OBJECTIVE

THE STUDENT WILL APPLY HIS KNOWLEDGE OF TRIGONOMETRY TOPROOFS AND PROBLEMS BY COMPLETING THE FOLLOWING LEVEL TOTHE SATISFACTION OF THE TEACHER.

I. Functions

Objective #1: The student will apply his knowledge oftrigonometric functions by completing thefollowing suggested activities to the sat-isfaction of the teacher.

Activities:


1. Define a function as a set of ordered pairs no twoof which have the same first element.

2. Recognize a function from both its equation and itsgraph.

3. Differentiate among an "into" function, an "onto"function, and a "one-to-one" function.

4. Define a radian.

5. Find the radian measure of a central angle of acircle given its arc length and the radius of thecircle.

6. Convert from radian measure to degree measure andvice versa.

7. Understand that the wrapping function ( 0 ) mapsthe real numbers onto the points of a unit circle.

8. Fina the points onto which certain real numbers,such as 0, m/3, 71/4, r/6, Tr/2, IT , 37/2, etc.are mapped.

9. Recognize the basic definitions of sin e and cos &.

10. Find sin 9 and cos 8 for real number values of 0 inall four quadrants.

11. Identify the definitions of tan 9, ctn e, sec 6, andcsc 0 in terms of sin 9 and cos 6.

12. Apply the basic relationships that exist among thesix circular functions.

M 108 p. 2

13. Recognize that the six circular functions are peri-odic in nature.

14. Describe the behavior of the circular functions overgiven intervals.

15. Graph the six circular functions with the followingmodifications:

a. Change of amplitude,

b. Change of period,

c. Horizontal translation,

d. Vertical translation,

e. Addition or subtraction of curves, and

f. Combinations of the above.

16. Apply the eight fundamental relationships that existamong the circular functions in proving identities.

17. Develop techniques which may be employed in the prov-ing of identities.

18. Apply the various sets of formulas which may be de-veloped from the basic definitions:

a. Addition formulas,

b. Subtraction formulas,

c. Double-angle formulas,

d. Half-angle formulas, and

e. Sum and difference formulas.

19. Identify the inverse of a function, in particular,the inverse circular functions.

20. Graph the inverse circular functions and recognizetheir ranges of principal values.

21. Evaluate expressions using the principal values ofthe inverse circular functions.

22. Solve open sentences involving the circular functions.

M 108 p. 3

II. Complex Numbers and Triangles

Objective #2: The student will apply his knowledge ofthe principles of real numbers and trig-onometry to problems involving complexnumbers and triangles by completing thefollowing suggested activities to thesatisfaction of the teacher.

Activities:


23. Express complex numbers in polar or trigonometricform.

24. Perform the operations of addition, subtraction,multiplication, and division of complex numbers inpolar form.

25. Solve equations using DeMoivre's Theorem.

26. Redefine the circular functions as trigonometricin terms of the sides of a triangle.

27. Solve for the various parts of a right triangle us-ing traditional methods and logarithms.

28. Solve for the various parts of an oblique triangleusing the Law of Sines or the Law of Cosines in thefollowing situations:

a. AAS,

b. SSA (Ambiguous Case),

c. SAS, and

d. SSS.

M 109 p. 1

LEVEL OBJECTIVE

THE STUDENT WILL INCREASE IN HIS ABILITY TO ANALYZE THE PRIN-CIPLES OF GEOMETRY BY COMPLETING THE FOLLOWING LEVEL TO THESATISFACTION OF THE TEACHER.

I. Points and Lines

Objective #1: The student will increase his ability toanalyze the properties of points and linesby completing the following suggested ac-tivities to the satisfaction of the teach-er.

Activities:


1. Set up a coordinate system on a line.

2. Set up a coordinate system on a plane.

3. Compute the distance between two points on a line.

4. Graph the solution set for the following:

a. x L a (a E R),

b. x_4. a (a 6 R),

c. = a (a 6 +R),

d. (xi 2' a (a 6 +R),

e. (xl a (a E +R), and

f. a b (a.< b, a, bE: +R)

5. Compute the distance between two points on a plane.

6. A 1 the sistance formula d = Y (x2 - x1)2 + (3,2

Y1)4 to various situations which may arise in ana-lytic geometry.

7. Compute the coordinates of a point which bisects agiven segment.

8. Determine the coordinates of the points which dividea given segment into n congruent segments.

M 109 p. 2

9. Identify the slope of a line as the tangent of itsangle of inclination.

10. Compute the slope of a line given two points throughwhich it passes.

11. Recognize the various forms of the equation of astraight line:

a. Two-point form,

b. Point-slope form,

c. Slope-intercept form,

d. Intercept form, and

e. Standard form.

12. Write the equation of a straight line in the abovefive forms given appropriate information.

13. Apply the fact that the slopes of parallel linesare equal to problem situations.

14. Apply the fact that the slopes of perpendicularlines are the negative reciprocals of each other toproblem situations.

15. Find the angle of intersection between two lines.

16. Determine theangles of a polygon knowing the co-ordinates of its vertices.

17. Compute the distance from a point to a line.

18. Apply the formula for the distance from a point toa line to various situations which arise in analyticgeometry.

19. Apply the methods of coordinate geometry to theproofs of theorems.

20. Apply the methods of coordinate geometry to find-ing certain loci of points.

II. Conic Sectinis

Objective #2. The student will increase his ability toanalyze conic sections by completing thefollowing suggested activities to thesatisfaction of the teacher.

M 109 p. 3

Activities:


21. Recognize that a circle is the locus of points in aplane at a given distance from a given point in theplane.

22. Determine the center and radius of a circle whenits equation is written in the following forms:

a. x2 + y2 = r2, and

b. (x - h)2 (y-

k)2 = r2.

23. Write the equation of a circle given various setsof information.

24. Define the eccentricity of a conic as a positivenumber which repre'sents the ratio of the distancefrom a point on the conic to a given focal pointto the distance from the same point on the conicto a given directrix line.

25. Recognize that the parabola is the conic sectionwith eccentricity e = 1.

26. Graph a parabola when its equation is written inone of the following forms:

a. x2 = 4py,

b. y2 = 4px,

c. (x - h)2 = 4p(y - k), and

d. (y - k)2 = 4p(x - h).

27. Recognize that the ellipse is the conic sectionwith e 1.

28. Graph an ellipse when its equation is written inone of the following forms:

a. x + = 1

a2 b2

b. + x2 = 1

a2 b2 a > b,

c. (x - h)2 + (y k)2 = 1, and

a2 b2

M 109

d. (y k)2 + (x - h)2 = 1.

a2 b2

29. Recognize that the hyperbola is the conic sectionwith e 1.

30. Graph a hyperbola when its equation is written inone of the following forms:

a. x2 - yl = 1,

a2 b2

b. y2_ - x2 = 1,

a2 b2

c. (x - h)2- (y - k)2 = 1, and

a2 b2

d. (y - k)2- (x - h)2 = 1.

a2 b2

31. Write the equations of the parabola, ellipse, andhyperbola given certain sets of information.

32. Find the coordinates of the points of intersectiorof various pairs of conics.

33. Recognize the equations of degenerate conics andgraph these special cases.

34. Recognize that the equation Ax2 + Bxy + Cy2 + Dx +Ey + F = 0 is the general equation of the second-degree.

35. Eleminate the cross-product (xy) term from the aboveequation by a rotation of axes.

36. Graph the transformed conic on the new set of ro-tated axes.

III. Parameters and Coordinates

Objective #3: The student will increase his ability toanalyze equations and the possible co-ordinate systems of graphing by complet-ing the following suggested activities tothe satisfaction of the teacher.

M 109 p. 5

Activities:

37. Recognize that a parameter is a third variablewhich relates the independent and dependent vari-ables, x and y.

38. Graph a curve from its parametric equations.

39. Eliminate the parameter and graph the resultingCartesian equation.

40. Determine the equivalency of the parametric andCartesian equations.

41. Construct a polar coordinate system.

42. Plot the points (r, 0) on a polar coordinate system.

43. Interchange the polar and Cartesian coordinates ofpoints.

44. Determine the symmetries of an equation when writ-ten in polar form.

45. Graph curves in polar from such as the following:

a. Cardioid,

b. Limacon,

c. Lemniscate,

d. Rose Curves,

e. Spiral, and

f. Lituus.

46. Transform an equation from Cartesian to polar formand vice versa.

47. Realize that some surves are graphed more easilyin Cartesian form and others more easily in polarform.

48. Compute the distance between two points when theircoordinates are given in polar form.

49. Determine the symmetries of a curve when its equa-tion is written in Cartesian form.

M 109 p. 6

50. Find the equations of the horizontal and verticalasymptotes of rational functions.

51. Graph rational functions with their appropriatehorizontal and vertical asymptotes.

M 110 p. 1

LEVEL OBJECTIVE

THE STUDENT WILL APPLY HIS KNOWLEDGE OF CALCULUS TO PROBLEMSOLVING SIUTATIONS BY COMPLETING THE FOLLOWING LEVEL TO THESATISFACTION OF THE TEACHER.

I. Functions

Objective #1: The student will increase his comprehen-sion of functions by completing the fol-lowing suggested activities to the sat-isfaction of the teacher.

Activities:


1. Recognize a polynomial function as one which has anequation y = °(x) . anxn + aixn - 1 + . .+an wheren is a positive integer and an, al, . . . an are re-al numbers.

2. Identify polynomial functions as constant, linear,quadratic, cubic, biquadratic, etc.

3. Find the domain and range of a polynomial function.

4. Understand the intuitive definition of a continuousfunction.

5. Compute the average rate of change of a function.

6. Recognize the average rate of change of a functionas the slope of a secant line to the curve.

7. Apply the concept of the average rate of change ofa function to practical situations.

8. Recognize the exact rate of change of a function asthe limit of the average rate of change as 41 x > O.

II. Derivatives

Objective #2: The student will apply his knowledge ofderivatives in calculus to the solutionof problems by completing the followingsuggested activities to the satisfactionof the teacher.

Activities:


M 110 p. 2

9. Identify the derivative of a function as its exactrate of change.

10. Identify the exact rate of change or the first der-ivative of the function as the slope of a tangentline to the curve.

11. Ccipute the first derivative of polynomial functions.

12. Write the equations of tangent lines to a curve atany point on the curve.

13. Recognize the average velocity as the average rateof change of the distance function.

14. Recognize the instantaneous velocity as the firstderivative of the distance function.

15. Recognize the average acceleration as the averagerate of change of the velocity function.

16. Recognize the instantnaeous acceleration as thefirst derivative of the velocity function or thesecond derivative of the distance function.

17. Apply the theory of distance, velocity, and accel-eration to practical problems.

18. Write the equations of normal lines to a curve atany point on the curve.

19. Differentiate between an increasing and a decreas-ing function.

20. Recognize points of relative maximum or minimum ona curve.

21. Find the critical values for a polynomial functionin x.

22. Find the maximum and minimum points on a curve byusing the first derivative test.

23. Find the maximum and minimum points on a curve byusing the second derivative test.

24. Apply maxima and minima theory to the graphing ofpolynomial functions.

25. Apply maxima and minima theory to the solution ofpractical problems.

M 110 P. 3

III. Integrals

Objective #3: The student will apply his knowledge ofthe concepts of integration to solvingproblems by completing the following sug-gested activities to the satisfaction ofthe teacher.

Activities:


26. Determine the antiderivative or integral of a poly-nomial function.

27. Apply integration to distance, velocity, and accel-eration problems.

28. Evaluate a definite integral.

29. Apply integration to the computation of areas.

LEVEL OBJECTIVE

THE STUDENT WILL APPLY HIS KNOWLEDGE OF THE CONCEPTS OFPROBABILITY AND MATHEMATICAL INDUCTION TO PROBLEMS BY COM-PLETING THE FOLLOWING LEVEL TO THE SATISFACTION OF THETEACHER.

I. Permutations and Combinations

Objective #1: The student will apply his knowledge ofthe concepts of permutations and combin-ations to problems by completing the fol-lowing suggested activities to the satis-faction of the teacher.

Activities:


1. Apply the fundamental prin-jole of counting inproblem situations.

2. Differnetiate between an "and" and an "or" connect-ive when using the fundamental principle of count-ing.

3. Realize that a permutation ;s the arranger-nt of agroup of objects in a definite order.

4. Use the following formulas in problem situations:

a. npn = n!,

b. nPr = n!

777)!, and

c. Repeated objects - n!

p!q!

5. Realize that a combination or a selection is agroup of things in which the arrangement or orderis not considered.

6. Use the formula nCr . n! in problem situ-r! (n - r)!

ations.

7. Differentiate h)tween permutation and combinationexamples.

M 111 p. 2

II. Probability

Objective #2: The student will apply his knowledge ofthe concept of probability to situationsinvolving mathematical expectation by com-pleting the following suggested activitiesto the satisfaction of the teacher.

Activities:


8. Realize that a probability is the ratio of the num-ber of ways an event can occur successfully to thetotal number of ways the event can occur.

9. Identify events that are mutually exclusive.

10. Compute the probability of mutually exclusive events.

11. Compute the probability of independent events.

III. Mathematical Induction

Objective #3: The student will apply his knowledge ofinductive reasoning to proofs by complet-ing the following suggested activities tothe satisfaction of the teacher,

Activities:

The student will be able t..:

12. Differentiate between the methods of inductive anddeductive thinking.

13. Define the concept of mathematical induction.

14. Realize that the two reguiremkts for mathematicalinduction are sufficient to prove the validity ofsentences.

15. Test the validity of sentences by the method ofmathematical induction.

M 112 p. 1

LEVEL OBJECTIVE

THE STUDENT WILL DEMONSTRATE AN INCREASE IN HIS ABILITY TOANALYZE THE REAL NUMBER SYSTEM AND ALGEBRAIC EXPRESSIONS TODEVELOP THE MATHEMATICS OF CALCULUS BY COMPLETING THE FOL-LOWING LEVEL TO THE SATISFACTION OF THE TEACHER.

I. Analytic Preparation

Objective #1: The student will increase his ability toanalyze geometric concepts by completingthe following suggested activities to thesatisfaction of the teacher.

Activities:


1. Review those topics of Analytic Geometry covered inthe Advanced Mathematics course.

2. Consider all definitions of the absolute value of x.

3. Consider geometric interpretations of x.

4. Analyze the triangle inequality and the absolute val-ue of sums, products, quotients, and differences.

5. Analyze the four kinds of intervals of real numbers.

6. Examine the definition for neighborhoods.

7. Analyze and apply the definition for a deleted nei-bo:hood.

II. Functions

Objective #2: The student will apply his knowledge ofthe concepts of functions by completingthe following suggested activities to thesatisfaction of the teacher.

Activities:


8. Investigate and use the definitions of a function.

9. Determine when two functions arl equal.

M 112 p. 2

10. Use the different methods of representing thefunctional relationship between the variables xand y which are:

a. by tables of values.

b. by corresponding scales.

c. by means of graphs.

d. by means of equations.

11. Recognize and graph the greatest integer function.

12. Recognize and graph the signium function.

13. Evaluate the composition of functions.

14. Investigate and study the behavior of functions.

15. Compute and graph the slope function.

16. Formulate the method for finding the derivative ofa function.

17. Investigate and solve problems regarding velocityand rates.

18. Ulerstand the definition of limit as an operationapplied to a function at a point.

19. Prove theorems about limits of sums, products, andquotients of functions.

20. Examine and apply the properties of infinity.

Derivatives of Algebraic Functions

Objective #3: The student will apply his knowledge ofthe derivative of a function by complet-ing the following suggested activitiesto the satisfaction of the teacher.

Activities:


21. Identify a polynomial in x as a function which isthe sum of a finite number of monomial terms.

22. Prove that the derivative, with respect to x, of xnis nxn-1 when n is any positive integer.

26. Apply the use of derivatives to mechanics by using

III

11

23. Prove that the derivative of a constant is zerpol

24. Prove that if u=f(x) is a differentiable function

25. Prove that the derivative of the sum of a finite

of x and c is a constant, then d (cu) c du

number of differentiable functions is equal to thesum of their derivatives.

M 112

dx = dx.

the first derivative to find the velocity of a mov-ing body and by using the second derivative to findacceleration.

27. Derive formulas for finding the derivative with re-spect to x of products (y=uv), quotients (y=u), andpowers (y=un). v

28. Calculate dy/dx from a method known as implicit dif-,ferentiation.

29. Estimate the change of y produced in a functiony=f(x) when x changes by a small amount x.

30. Derive and use the chain rule for derivatives.

31. Adopt the definitions of dx and dy in differentialnotation.

32. Use the geometric interpretation of dy and dx.

33. Use differentials to obtain reasonable approximations.

34. Recognize the continuity of a function.

35. Consider how the sign of the derivative at a pointgives information about the curve.

36. Solve problems involving related rates.

37. Examine the significance of the sign of the secondderivative.

38. Follow the procedure for graphing the equation y=f(x).

39. Analyze the theory regarding maxima and minima.

40. Solve problems involving maxima and minima.

M 112

41. Derive and apply Rolle's Theorem.

42. Prove and apply the Mean Value Theorem.

IV. Integration

p. 4

Objective #4: The student will apply his knowledge ofthe concept of integration by completingthe following suggested activities to thesatisfaction of the teacher.

Activities:


43. Analyze the meaning of integration.

44. Solve differential equations.

45. Find a function by use of the indefinite integral.

46. Con ider the applications of indefinite integration.

47. Review formulas and theory from trigonometry.

48. Prove the theorem lini sui 2

2-0 2 = 1.

49. Derive the derivatives and integerals of y=sin u

and y=cos u.

50. Find the area under a curve.

51. Compute areas as limits.

52. Find areas by calculus.

53. Derive and apply the definite integral and theFundamental Theorem of Integral Calculus.

54. Approximate an integral by using the trapexoidalrule.

55. Use summation notation.

56. Find the area between two curves.

57. Calculate the distance traveled by a body movingwith velocity v=f(t).

M 112 p. 5

58. Solve problems regarding the volume of a solid ofrevolution by the disc method and by cylindricalshells.

59. Find the length of a plane curve.

60. Solve problems regarding the area of a surface ofrevolution.

61. Find the average value of a function.

62. Solve problems regarding centroid and center ofgravity.

63. Solve problems regarding moments and center of mass.

64. Solve work problems.

V. Transcendental Functions

Objective #5: The student will increase his ability toanalyze trancendental functions by com-pleting the following suggested activ-ities to the satisfaction of the teacher.

Activities:


65. Study the meaning of the word transcendental.

66. Derive the other trigonometric functions not previ-ously derived.

67. Examine and review the inverse trigonometric func-tions.

68. Analyze thenatural logarithm of x.

69. Derive the derivative of In x.

70. Establish the properties of the natural logarithms.

71. Graph y=ln x.

72. Analyze the exponential function.

73. Analyze the function au.

74. Establish the properties for the function au.

M 112

75. Analyze the function logau.

VI. Mehtods of Integration

p. 6

Objective #6: The student will increase his ability toanalyze various methods of integration bycompleting the following suggested activ-ities to the satisfaction of the teacher.

Activities:


76. Review the basic formulas.

77. Learn and apply the formula for powers of trigon-ometric functions.

78. Examine and apply the method used for even powersof sines and cosines.

79. Analyze and apply the method used for integrals in-volving the square root of a2-u2, square root ofa2+u2, square root of u2 -a2, a2+u2, and a2-u2.

80. Analyze and apply the method used for integrals in-volving ax2+bx +c.

81. Integrate by the method of partial fractions.

82. Integrate by parts.

83. Integrate rational functions of sin x and cos x andother trigonometric integrals.

84. Integrate by using further substitutions.

85. Investigate improper integrals and their conver-gence.

86. Use Simpson's Rule for approximation.

VII. Plane Analytic Geometry

Objective #7: The student will increase his ability toanalyze geometry by completing the follow-ing suggested activities to the satisfac-tion of the teacher.

Activities:


M 112 p. 7

87. Review those topics of Analytic Geometry coveredfrom the Advanced Mathematics course.

88. Graph higher degree equations using horizontal andvertical asymptotes.

89. Find equations of the tangent and normal lines.

90. Review those topics of Polar Coordinates coveredfrom the Advanced Mathematics course.

91. Review the graphs of polar equations.

VIII. Hyperbolic Functions

Objective #8: The student will increase his ability toanalyze hyperbolic functions by complet-ing the following suggested activities tothe satisfaction of the teacher.

Activities:


92. Analyze the meaning of hyperbolic functions.

93. Compare hyperbolic functions with circular functions.

94. Derive the equations of the hyperbolic sine and co-sine.

95. Derive the remaining hyperbolic functions in termsof the sinh u and cosh u.

96. Graph the hyperbolic functions.

97. Establish the derivatives and integrals for thehyperbolic functions.

98. Establish the geometric significance of the hyper-bolic radian.

99. Analyze the inverse hyperbolic functions.

100. Establish the derivatives and integrals for the in-verse hyperbolic functions.

101. Derive the equation of the iv:Aging cable.

102. Define and sketch the curve referred to as the gud-ermaninan of x.

M 112 p. 8

IX. Vectors and Parametric ::quations

Objective #9: The student will increase his ability toanalyze vectors and parametric equationsby completing the following suggested ac-tivities to the satisfaction of the tea-cher.

Activities:


103. Analyze parametric equations in kinematics.

104. Examine the use of parametric equations in analyticgeometry.

105. Analyze the use of vector components and the unitvectors i and j.

106. Differentiate vectors.

107. Examine a unit vector tangent to the curve at P.

108. Explore curvature and normal vectors.

109. Examine tangential and normal components of thevelocity and acceleration vectors.

110. Investigate the particle P moving on a curve whoseequation is given in polar coordinates.

document resume se 016 300 secondary schools curriculum ... · behavioral objectives for geometry,...

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